Alright, so you see uhhh what we did last class was uhhh uhhh to define the notion of a morphism uhhh uhhh and the idea of a morphism is that uhhh it is a continuous map which should pull back regular functions to regular functions okay that is the uhhh that is the definition of a morphism and then uhhh I stated uhhh uhhh I stated a few results let me recall them, so uhhh let uhhh x uhhh be any variety any variety in our case uhhh it means either it is uhhh an affine variety or quasi-affine variety an affine variety as you know is an irreducible closed subset of affine space and we are working over an algebraically closed field okay and a quasi-affine variety is a non-empty open subset of an affine variety and uhhh variety for us at the moment means either an affine variety or a quasi-affine variety which means either an irreducible closed subset of affine space or an non-empty open subset of an irreducible closed subset of affine space and uhhh we saw we saw that uhhh uhhh if x is affine okay then uhhh the uhhh then a x and o x are isomorphic okay. So a x is the uhhh the coordinate ring of x uhhh the ring of uhhh uhhh the ring of polynomial functions on x which is gotten by taking the ring of all polynomial functions of the affine space in which x is sitting inside modulo the ideal uhhh of x which is a prime ideal and o x is supposed to be the ring of all uhhh regular functions on x okay and uhhh we we saw that if x is an affine variety then there is an isomorphism like this alright. And uhhh mind you uhhh this isomorphism is not just an isomorphism of rings it is an isomorphism of k algebra okay it is an isomorphism of rings which is also an isomorphism of k vector spaces okay so k algebra isomorphism and uhhh of course uhhh uhhh o x was defined to be the ring of regular functions on x and uhhh regular functions were defined to be functions which were locally given by quotients of polynomials okay. Now we then uhhh I told you that uhhh the the uhhh we defined uhhh then the notion of a morphism of one variety into another variety and what we did was uhhh uhhh we showed the set of morphisms of varieties from x to a1 uhhh or let me put y to a1 is isomorphic uhhh uhhh or let me let me again put x uhhh or let me put y okay y to a1 is isomorphic it can be identified with oi uhhh for any variety for any variety y namely uhhh every regular function which is a map from y to k okay is is the same as a morphism from y to a1 where k is thought of as a1 uhhh which means that a1 is just k with these are zerski topology and the zerski topology there of course is you know it is a complement finite topology is a topology for which the open sets are complements of finite sets the closed sets are only finite sets okay uhhh. So and I told you that this uhhh uhhh uhhh this can be generalized that the statement can be generalized uhhh so you know this statement can be written as morphisms of varieties from uhhh uhhh x so let me let me go back to uhhh uhhh so let me again write y uhhh to a1 is isomorphic to this can also be written as morphism of uhhh k algebras from uhhh o of a1 k to o of 1 because this o of a1 k is actually isomorphic to kx o of a1 is just the uhhh uhhh o of a1 is the same as a of a1 okay and uhhh uhhh and a of a1 is k of x it is a polynomial ring in one variable okay. So if you want uhhh if you want I can write uhhh I am I am I am using the fact that you know uhhh o and a are the same for a fine variety so if you want I will write this is this can be identified with a of a1 and uhhh uhhh and that is uhhh that is isomorphic to kx okay. Then the all possible k algebra homomorphisms from the polynomial ring kx to any any k algebra is in bijection with the elements of that k algebra okay because this is the universal property of the polynomial ring in one variable okay namely any k algebra homomorphism from kx to o of y is completely controlled by what by the image of x okay it is controlled by the element of o of y that x goes to okay and therefore you will have as many k algebra homomorphisms as there are elements in o of y. So this set is the same as o of y okay so I am just translating the statement like this using the universal property of the polynomial ring in one variable and then I told you that uhhh more generally you can also write it as uhhh what I have written for a1 you can write it for an and not only for an you can write it also for an any affine variety okay. So uhhh this statement is morphism of varieties from y to an can be identified naturally with morphisms of k algebras uhhh from o of a from uhhh a of an to o of y uhhh and uhhh what I can do for a1 uhhh uhhh and I have done for an I can do it also for an irreducible close subset of an namely an affine variety. So I can also write it as morphism of varieties y, x is isomorphic to is natural is in a natural bijective correspondence of morphisms of k algebras from a of x to o of y okay where of course x is uhhh where x is of course an affine variety okay. So this is the most general statement this is the most general statement okay. So in this general statement if you put x equal to an you get this statement if you put x equal to a1 you get this statement which actually so if you put x equal to a1 it boils down to this statement that regular functions are nothing but morphisms into a1 okay regular functions are no different from morphisms into a1 right. So what I am going to do is uhhh this is a very very important statement okay because as I told you at the end of the previous lecture that this is the statement that gives an equivalence of categories between uhhh on the one side the category of affine of affine varieties with morphisms being morphisms of affine varieties on and on the other side the category being the category of uhhh finitely generated k algebras that are integral domains with morphisms being k algebra homomorphism. So what I am going to do is I am really going to uhhh focus on proving this fact okay. So that is what I am going to do now it is a very important fact. So uhhh uhhh proof of uhhh uhhh proof of the statement morphisms let me rewrite it again uhhh varieties from y to x isomorphic uhhh here of course this means uhhh there is a bijection of sets okay is this what you have here is a set what you have there is a set. But the point is that this side represents the geometric side the algebraic geometric side this side represents the uhhh commutative algebra side okay and as usual uhhh whole point about uhhh algebraic geometry is that you go from the uhhh you know from the geometric side to the commutative algebra side uhhh this translation is very important to algebraic geometry okay. So what I am trying to do is I am trying to prove this uhhh this is the algebraic geometric side morphism of varieties from y to x and here is a commutative algebra side which is morphisms as k-algebra of uhhh A y A x to O y. So this is this is what I am going to prove okay. So uhhh I am going to prove that there is a bijection like this there is a natural bijection like this. Now how do I do that? So so define alpha from morphism so again let me write this down I define a map like this first and a map like this which is uhhh if you give me a morphism uhhh phi of varieties from y to x this is an element on this side I have to send it to something here and you know uhhh uhhh what I am going to send it to here is uhhh a very natural thing it is it is a thing that characterizes what a morphism is a morphism is a continuous map which is supposed to be uhhh as uhhh as I considered as a pull back map something that pulls back regular functions to regular functions. So what I am going to do is I am going to just take the uhhh pull back pull back via phi of regular functions and what does that mean it means so the notation for that is phi upper star which is a map from regular functions on y to regular functions on x okay. So this is my map so uhhh what is so what is it that is what is it that is happening if you give me uhhh if you give me a regular function uhhh f uhhh on y I get uhhh the pull back of that regular function to x is a regular function on x uhhh this is the defining property of a morphism apart from its continuity and what does this mean this is just composition. So you know you have you have y uhhh to x you have this map uhhh you have this morphism phi and uhhh f is a regular function on y uhhh I have uhhh yeah I have messed up x I have I have mixed up x and y this should be x and this should be y. So you have a regular function on x f is a regular function on the target so it is uhhh if you want f is a morphism into a1 uhhh or it is a regular function into k okay and then uhhh what is the pull back it is this map which is a composition this is first apply phi then apply f okay this is uhhh this is a this is otherwise called as phi upper star of f this is the pull back and the fact is that if you have a regular function on the target uhhh by composing with the morphism you should get a regular function on the source the target is x so give me a regular function on x I am by composing with phi the pull back will be a regular function on y this is the definition of a morphism okay this is part of the definition of a morphism and mind you that this o x can be this o x can be uhhh uhhh can is it can be canonically naturally identified with a x in fact I should even put equality okay o x is the same as a x alright uhhh so here also when I say that if x is affine then a x and o x are isomorphic uhhh in fact you can say that the in fact they are equal okay if you think of them as functions into k uhhh with into a1 with which is k with this k topology they are equal actually this is actually an equality okay and so it is very clear that I have a very natural map given any morphism look at the pull back map which takes regular functions to regular functions and it is very easy to see that uhhh this pull back map is of course a k algebra homomorphism because you know phi upper star of f f1 plus f2 will be phi upper star of f1 plus phi upper star of f2 and uhhh phi upper star of f1 into f2 will be phi upper star of f1 into phi upper star of f2 you can easily verify that this what you get here is not just a map from a x to o y but that is a k algebra homomorphism because functions by the definition of functions uhhh everything is done point wise some of two functions uhhh is defined point wise uhhh product of two functions is defined point wise multiplying a function by a scalar is a new function which is also defined point wise. So you can easily check that this is a this is certainly a k algebra homomorphism okay what we need to prove is that this map uhhh so uhhh I have written alpha but uhhh well so this is alpha of phi but the better notation is phi upper star okay uhhh but since I have written alpha here I have also write it here alright and what will I have what will I have to prove I will have to prove that this map is injective I have to prove this map is adjective that is all I have to do alright. So let us uhhh let us try to prove the injectivity let us try to prove this injectivity. So uhhh I guess the injectivity is very easy so uhhh so what is the situation uhhh suppose there are uhhh suppose I have two morphisms which give rise to the same k algebra homomorphism uhhh upon as pull backs of regular functions then I have to say that these two are the same okay. So suppose uhhh alpha of phi 1 is equal to alpha of phi 2 which is the same as saying that you know phi 1 upper star the same as phi 2 upper star okay I will have to show phi 1 is the same as phi 2 right we need to show uhhh uhhh we need to show that phi 1 is the same as phi 2. So you have to show that these two morphisms are one and the same and what do you mean by equality of morphisms by equality of morphisms means equality of the underlying maps as sets okay equality is just a set theoretic maps. So it is a set theoretic checking that you have to uhhh do okay. So uhhh so how do I check two maps are equal I check two maps are equal by evaluating them at every point and showing that they are equal. So what I do is uhhh well so here is my situation. So I have so I have why uhhh so there are these two maps phi 1 and phi 2 which are morphisms into x and mind you x is an affine variety x is an affine variety which means that uhhh uhhh x is sitting inside x is an irreducible go subset of a suitable a n.  So the when I say a n with coordinates t 1 etc t n what it means is that I am just saying that if you take the if you take the uhhh ring of uhhh polynomial functions on a n I am identifying it with k polynomial ring in t 1 etc t n okay this is what it means. The choice of variables in the ring of polynomials in a n those variables uhhh the choice of variables will also give you the uhhh labels for the coordinate functions okay and uhhh uhhh what happens is that you know this is a this is an irreducible closed subset therefore uhhh uhhh this if I go mod the ideal of x which is going to be a prime ideal I am going to get a of x okay. So it is going to be uhhh uhhh so this is the this is the polynomial functions on a n restricted to x okay uhhh to get the ring of functions on x I will have to go modulo the uhhh ideal of x which means I will have to identify two functions on x uhhh if and only if they differ by uhhh a function that is 0 on x and that identification corresponds to going modulo I of x okay uhhh and uhhh so you know under this uhhh uhhh under this map uhhh you know all the every xi any any ti will go to some ti bar okay if you want uhhh thought of as uhhh an element here okay and mind you ti bar is uhhh geometrically ti bar is just the coordinate function ti restricted to x. So this ti bar is just ti restricted to x okay actually what is this quotient map given any polynomial uhhh given any polynomial consider it as a function on the affine space and restricted as a function on x that is this quotient this restrict this quotient map is actually restriction geometrically it is restriction of polynomials to a closed subset this closed subset x okay. And uhhh uhhh how do I check these two maps are equal I check uhhh these two maps are equal by uhhh looking at the image of a point. So let me take a point small y okay the point small y will go to uhhh under phi 1 it will go to phi 1 of y uhhh uhhh and this is a point it is going to be a point of x and under phi 2 it is going to go to phi 2 of y which is a point of x again okay and both phi 1 of y and phi 2 of y are going to be points in affine space. So they are going to have coordinates so uhhh let uhhh phi 1 of y be well uhhh lambda 1 etc lambda n because it is a after all phi 1 of y lies in x and after all it is a point in n space it is given by n coordinates okay and phi 2 of y is say given by mu 1 etc mu n these are the coordinates of this point okay. And what I will have to prove is that if phi 1 star phi 1 upper star equal to phi 2 upper star then I will have to show that these lambdas are the same as the mu's which will say that the point phi 1 of y and phi 2 of y are one in the same point and if I and since y was arbitrary it would tell you that phi 1 and phi 2 are one in the same map okay. So uhhh so the uhhh the the the point that one has to understand is that uhhh what is the meaning of uhhh phi 1 upper star equal to phi 2 upper star. So, phi 1 upper star is the pull back map from the ring of functions on the target the regular functions on the target to the regular functions on the source. So, it is from A x to A y uhhh 2 of y of course A x is the same as O x because x is affine okay. So, the pull the this is the pull back okay. And what does the pull back map do uhhh given a function uhhh given a regular function on x I compose it with this to get a regular function on the source uhhh that is how the pull back map works. Take a regular function on the target compose it with the map you get a regular function on the source. So, uhhh so in particular you know the the ti's okay restricted to x they are also regular functions on the target. Mind you A x is A x is uhhh the same as O x because x is affine. So, this ti bar they are regular functions on x the coordinate functions they are just the coordinate functions on the affine space restricted to the irreducible cross subsets okay. So, if I take each of this ti bars which are which is the same as ti restricted to x they will go to this it will go to phi 1 upper star of ti okay. And uhhh what is given to me what is given to me is phi 1 upper star is the same as phi 2 upper star. So, what it what it means is phi 1 upper star equal to phi 2 upper star means that this is the same as phi 2 upper star of ti bar for every r this is what it means this is what is given to me okay. Now just think for a moment what is the meaning of phi 1 upper star of ti bar phi 1 upper star of ti bar is the regular function on y which is gotten by pulling back the regular function ti bar on x. And what does what does pull back do pull back is a composition with phi.  So, if you translate that you will get t uhhh ti bar composition phi 1 is equal to ti bar composition phi 2 this is true for every i this is what it means now now this is an equality of regular functions on y. So, apply it to the point small y. So, what you will get is ti bar of phi 1 of small y is equal to ti bar of phi 2 of small y. So, this will tell you that ti bar of phi 1 y is equal to ti bar of phi 2 y okay. But phi 1 y is uhhh uhhh uhhh phi 1 y has these lambdas as the coordinates, phi 2 y has mu's as the coordinates. And if you take a point like this the action of ti bar on that point is to give you the ith coordinate okay it is just ti the function ti on affine space picks out the ith coordinate of the point when evaluated at a point. So, if you do that what I am going to get is that I am going to get lambda i is equal to mu i and this is going to be true for all i. And this implies that uhhh phi 1 of y is equal to phi 2 for all y in y and that tells you that phi 1 equal to phi 2 okay. So, uhhh so if alpha of phi 1 equal to alpha of phi 2 then you get phi 1 equal to phi 2 and that is injectivity okay okay. The slightly more uhhh involved uhhh part of the proof is a surjectivity namely to show that if you give me a k algebra homomorphism from A x to O y it arises from a morphism okay. And of course when once you prove it arises from a morphism we are already saying that that morphism is unique because we already proved uniqueness that is the injectivity okay. So, uhhh surjectivity is what we have to see next so how does one do the surjectivity. So, uhhh so what I am going to do is I am going to start with so here is my situation uhhh I have these morphisms of varieties from y to x I have this map alpha and on this side I have morphisms of k algebras from A x which is the same as O x to O y okay. And I am trying to prove surjectivity of this map I have already proved injectivity so what I do is I start with I start with uhhh uhhh uhhh with the k algebra homomorphism zeta okay. So, zeta I start with an element here zeta is a k algebra homomorphism from A x to O y I start with the zeta and I am trying to show that there exists a phi such that alpha of phi namely phi upper star is the same as this zeta this is what I have to prove okay. So, so but but what is A x A x is if you look at it carefully A x is just k t1 tn mod I x okay and so this A x is just k t1 etc tn mod I x okay this is what it is. In fact uhhh so it is k t1 bar etc tn bar it is it is generated uhhh by the images of the ti's okay it is it consists of just polynomials in the ti bars that is all okay.  See after all uhhh the ti bars are regular functions on x okay and zeta is a ring homomorphism it is a k algebra homomorphism. So, it is going to take an element here to an element here. So, if I take ti bar it is going to give me an element there and what is an element there it is a regular function on y. So, f i is the image under zeta of the ti bar for every i okay. So, the modular story is very simple the modular story is the moment you give me a k algebra homomorphism from k A x to O y the n coordinate functions on A x which are actually the n coordinate functions in the affine space in which x is sitting restricted to that uhhh x. They automatically give you a bunch of n functions on O y okay what does what does it mean to give you n regular functions on O y it means that you are giving n maps uhhh into k n scalar maps into k on y but that means what you are trying to do is if you think about it for every point of y if you evaluate this n maps you are getting an n tuple of points okay. So, you can see that this should give you a map from y to a point to an and that will be the map that will be that will be the morphism that you are looking for okay that is how you get this ejectivity. So, we will have to use these f i's to define a map from y to an okay by simply evaluating a point of y at uhhh these f i's and putting them as coordinates okay. So, you will get a map from y to an and then we will easily see that the map will actually factor through x and we will prove that the map is a morphism and we will prove that the alpha of that map is zeta okay therefore we will get this ejectivity of zeta okay. So, so let us do that. So, what I am going to do is let me rub this side off so I mean what you should remember is uhhh that the the moment you are given uhhh uhhh n regular functions on a variety you must always remember this whenever you are given n regular functions of on a variety you are actually uhhh given a map of that variety into an this is the idea okay. So, what so let me write this down define uhhh phi or let me use psi from y to an by y going to f 1 y f n y okay this is a this is a this is a well defined map because the f i's are all regular functions on y. If you evaluate a point of y at any of the f i's we are going to get a scalar you are going to get an element of k and this is an n triple of elements of k. So, it is a point in an which is just k n with a rescue topology and uhhh here is my map alright. Now, the first claim is is that uhhh psi uhhh maps into the first claim is that psi maps into x this map is not just going into an but it is going to go into the irreducible closed subset x of an it is going to go into it is going to go into uhhh this irreducible closed subset that is because it came from zeta as you will see. So, how do I show psi maps into x? So, it is very simple uhhh so how do you so what does it mean to uhhh how how does one verify this you have to show the give a point of given a point of small y alright given a point small y of capital Y the corresponding point psi of small y is a point in of an which actually lies in x this is what you have to show but how do you show a point of an find space lies in irreducible closed subset you will show that by showing that that point is in the common zero locus of the ideal of x how do you show a point of an lies in uhhh closed subset you will just have to show that it is a common zero of the ideal of that closed subset. So, all I have to do is I would take a point small y in capital Y I will take its image under psi which is this okay and then I have to verify that this point the point with this coordinate these coordinates satisfies every uhhh equation every polynomial in the ideal of x okay. So, that is what I have to do so so let uhhh g belong to uhhh ideal of x okay let g be in the ideal of x mind you g is a polynomial in n variables t1 tn g is a polynomial in so many variables okay then uhhh g restricted to x is actually g bar is is 0 in ax which is just k t1 etc tn by ix obviously an element in the ideal if you take its image in the quotient it is 0 but then you see uhhh from ax I have this map zeta I have started with the map zeta which is a k algebra homomorphism from ax to o y okay zeta is what I started with and zeta is a k algebra homomorphism in particular it is a ring homomorphism and ring homomorphism will take 0 to 0 okay. So, moral of the story is that if I take 0 which is g bar under if I apply zeta I will get zeta of g bar is 0 in o y this is what it means okay.  So this is certainly a regular function and it is 0 in o y so that means if I evaluate it at every point of capital Y I am going to get 0 so this will tell you that g of f1 of y etc fn of y is 0 for all y small y in capital Y this is what it means a regular function is 0 means if you evaluate it at every point it has to be 0 but what does this mean this is this is the same as saying that the point f1 y with coordinates f1 y through fn y this point is actually in the 0 set of g okay. So what I have proved is but what is this point this is just psi of y I have proved that for every small y in capital Y the corresponding point psi y is a 0 of every function g which is an ideal of x and that tells you that psi lands in x okay. So therefore this map that you got from y to a n actually factors through the closed subset x so you actually have a map into x okay. Now the next part of the proof is to show that this map is a morphism and I will have to show that this map is a morphism and I have to show that the pullback map that it induces is actually your zeta and then I would be done alright so let us see how to do that that is also pretty easy to do okay. So here is an exclaim the exclaim is psi is a continuous map the exclaim psi is a continuous map okay so situation is like this I have y I have psi this goes into x which is sitting inside a n okay and I want to show that this is continuous okay and mind you if I show that it is continuous I am actually more or less done showing that psi is a morphism because the movement psi becomes continuous okay then the only thing I have to check to check that psi is a morphism is that it pulls back regular functions so regular functions but the way we have defined it it will follow very trivially that zeta is actually psi star that is alpha of psi that will be very simple to see okay. So actually this is the crucial thing that needs to be verified that it is continuous okay. So the well the to see that that is also pretty easy let me think for a moment yeah so let me so it probably involves showing that the pullback of the pullback under psi is actually zeta I need that. So let me do that so let me write that here psi induces zeta and psi is a continuous map okay how do I explain that psi induces zeta it is very very simple. So you see alpha of psi which is psi star is going to be a map from ax to oi okay. In fact I should not say oi you know strictly I should say I can define for any map from if psi is any set theoretic map from y to x then given any set theoretic function on x I can pull it back to give a set theoretic function on y. So you know what I will do is I will just write maps from y to k okay and what is this is the usual pullback map pullback map exists for any set theoretic map even at the set theoretic level okay. So you see if you give me if you give me any polynomial h if you give me any element h which is a polynomial restricted to x actually then h goes to psi upper star of h and what is psi upper star of h is just first apply psi and then apply h this is the map okay and what I want you to understand is that what is this map see if you take a point of y h circle psi of y is h of psi y okay this is by definition psi y has been given a definition this is just h of f1 y and so on fn y because that is what psi y is okay but this is if you look at it h of f1 y fn y if you look at this calculation it is actually zeta of h operating on y because you see zeta of g is g of f1 etc fn so similarly h of f1 etc fn is zeta of h by the same argument. So moral of the story is so this will tell you that psi star it will tell you that and mind you so this will tell you that psi star is actually zeta and it will also tell you that psi star goes into goes just not into maps from y to k it goes into regular functions on y o of y is a subset of this is a set of all possible maps from y to k okay just all possible functions they need not even be continuous okay whereas o y is very special these are all morphisms of y into k they are the regular functions on y they are very special okay and the fact that psi star is zeta which you have verified point wise will tell you that and since zeta takes values in o y it tells you that psi star also lands inside not just in this but in this subset okay. So this proves that psi induces zeta okay so the only thing to show that the zeta is started with comes from a psi here is to show that actually psi is here namely I should show that psi is actually a morphism okay and what is the defining property of a morphism the defining property of a morphism there are two conditions the first condition is it has to be continuous the second thing is it should pull back regular functions to regular functions the second condition is already satisfied the pull back map induced by psi is already zeta and zeta is pulling back regular functions to regular functions it is taking every regular function on x to a regular function on y okay. So it is already pulling back regular functions to regular functions therefore the only thing that one has to check is actually that it is a continuous map okay so that is the reason why I said in the beginning that the essential claim is that it is a continuous map and once you do that you have proved everything you have proved it here alright. So how do I check something in the continuous map I will have to just check that you know inverse image of open sets are open or I have to check inverse image of closed sets are closed but that is a pretty easy thing because you see how do you define the Zariski topology on a on x there what is the Zariski topology on x how do you define the Zariski topology you for example you specify closed subsets given as zeros of polynomial okay. So what is a closed subset of x it is given as a common zeros of bunch of polynomials okay and if you compose those polynomials with psi okay those polynomials are regular functions okay therefore if you compose them with psi the resulting things will be regular functions on y and the inverse image will be precisely the locus of those zeroes of those regular functions on y and what you must understand is the Zariski topology given by the Zariski topology on y coincides with the topology that you get by taking the closed subsets to be common zeros of regular functions you see after all either you can when we started defining the Zariski topology we always took set of common zeros of polynomials but then when we went to a general variety which is which could even be an open subset of a closed irreducible subset then we had to define regular functions and these regular functions were locally given by quotients of polynomials okay and the zero set will be therefore locally the zero set of the numerator polynomial which is anyway closed okay it has got nothing to do with the denominator polynomial okay the zero set of a quotient of polynomials is essentially the zero set of the numerator the denominator polynomial is anyway assumed to be non-zero okay otherwise we would not put it in the denominator.  So what this argument tells you is that psi pulls back closed sets to closed sets so it is continuous and we are done okay. So let me write that down psi is continuous as psi inverse of z of some j is going to be psi inverse of is going to be intersection of okay so if you want I can write the full-blown set theoretic thing that you might so z of j is h inverse of zero intersection where h belongs to j right what is a zero set of an ideal zero set is you take this for each element in the ideal each polynomial in the ideal look at it zero which is h inverse of zero and you take the intersection this is the common zero set and you know psi inverse taking inverse is behaves well with respect to intersection so this is going to be intersection h belongs to j psi inverse of h inverse of zero okay and if you see that this is just the zero set in this is just the zero set of first apply this is a zero set of h circle psi intersection of the zero set of h circle psi where h belongs to j okay and this is closed closed in y so this implies psi is continuous and since psi is continuous it is already the map that induces pulls back regular functions to regular functions so it is a morphism and so I have found psi here I have been able to find psi I have been able to find psi which goes to z that gives me the physicality and I am done with the proof okay so proof ends here. So the crucial point that you will have to understand is the following which you can think of as you know as a separate exercise if you want I mean it is actually if you understand it if you have understood it well it is just a tautology but take any variety take a bunch of regular functions okay then the common set of common zeros of those regular functions is a closed subset of the variety okay it is a closed subset because regular functions are locally quotients of polynomials and taking the common zeros of such regular functions actually amounts to taking common zeros of a bunch of polynomials namely the numerator polynomials locally okay and that is the fact that you will have to understand okay so I will stop here.


Alright guys, so this is going to be a very quick and easy tutorial for you to follow if you have any questions Please let me know. I did do another video tutorial on this previously for my truck I had one that said back off and it has currently over 50,000 views and counting So I want another a little tiny Car detail on my car and I wanted to show you quickly how to do this So I'm choosing one that says seven dust if you don't know who seven dust is they are my favorite band So go check them out if you're interested in it. They are heavy metal, but that is what I love So that is my favorite band seven dust This is what it's going to look like, but I wanted to show you exactly how to do this So I'm going to be starting in Google here So I'm gonna pull it up and I'm gonna start in Google and I'm just gonna type in seven dust SVG and you can use anything it doesn't have to be seven dust you can use You know, whatever band you like you can do an animal that you like you could do whatever you want any kind of Decale you want on my car. I did one for my daughter for a band that she loves and everybody was asking me to create her one So it was kind of neat. So again, just starting in Google Type in whatever you want SVG and then I click on images and then I look through them and figure out which ones I can use so some of them you have to pay for oops. I'm trying to drag this down, but it doesn't want to All right, so we'll just skip that so I'm I'm trying to find one that has a white background or that doesn't say You know a watermark on it or anything like that. So I found one. It was just at the top here I chose this one and then I chose this one. So what I did was I right-click with my mouse And then I go to save image as and then you save it whatever you want it to be called I'm just gonna leave it like this and I'm gonna click save and I'm gonna replace it because I already have one saved in Here then I'm going to use this one So I'm gonna right-click do the same thing same image save image as and then it's gonna pull it up And I'm gonna change the name on this one to two because I don't want to replace the one that I just did So now it is up here in my downloads I'm gonna shrink my screen a little bit and show you guys how to do this and Unfortunately, I think I have to shrink this screen too Just because I have two screens and I don't think it's going to Pull it up on here. The reason I'm pulling up two screens is so you could see me So I'm gonna go to upload here and you guys might already know how to do this But in case you don't I'm just gonna show you really quick So you're gonna go to upload images and this is why I wanted to shrink my screen I'm gonna take my download here and all I do is click it and drag it in here see and there it is I'm gonna click on simple I'm gonna click continue and all I want to do is make sure to get rid of all the white spaces there so that it just Has these checkerboard on here. It's exactly how I want it I'm gonna click apply and continue I'm going to cut I'm going to click on the cut image because I don't want to print and cut it I'm gonna click upload upload. I'm gonna do it again for the next one So I'm gonna go to upload image go back to my little download section here and then get the second one So I know it's index here. It is again simple push continue and then just get rid of the gray or white spots Do you have what is left here? I'm gonna click continue I'm gonna go here to cut image and upload and make my screen bigger so we can not see all the stuff behind it And now I'm just going to go to Let's see. We're just gonna click upload again to get rid of that In there. So here's what I have. I'm actually sorry guys.  I want it centered and I wanted it the different size So here it is the seven dust one here looks a little bit different than this because I found it somewhere else But just to visualize what it's gonna look like and show you how to do it. This is what it is So I'm just gonna grab this make it larger and like I said, it doesn't have to be this It can be anything you want actually did a few of these videos So if you're interested in learning more or finding out how to make more of these I do have a couple of videos. Like I said, I had that really big one the back off one that I did in the back of my truck window Again, 50,000 views. Thank you guys for all of that. I really appreciate all the views and comments on that So all I did with this was just make it a little bit bigger You can unlock it here and change it to whatever size you want, you know, I can leave this one down here I can make a little bit bigger But basically all I did was that that's it I just change the size and then I select both of the images and I go up to align and I center Center it which was pretty good. So there it is Um, I'm actually you know what I like both of these I think I might cut out both of these. So one last thing I did for this was I actually pushed weld on here Just in case the pieces were not attached together in here I pushed weld instead of attached because sometimes if it's touching right here or you know, right here Even cricket might cut it out like a line here. So that's just why I do attach.  You know, whatever, but I just I I like to show people Things so I'm gonna make my actually a little bit bigger Not really I'm not paying attention up here You can do it the width and height that you want like if you wanted a specific size I don't really mind.  So I'll show you what it looks like when I'm finished So now that I'm done with that. I know it's both on basic cut. I do want them both on actually, you know What I'm thinking about I do want them white So I'm gonna click actually both of these go up to the color box here and change it to white Just because I want mine to cut out white because my windows are tinted black So this shows it a little bit better here. So if I did not weld this right here, I wish I could unweld it One thing if you push weld, you cannot undo it So be careful when you do it because once it's done it is finished Again, if you don't push attach cricket will actually cut little pieces out on each one of these You want to make sure you push weld.  I do want it to be permanent I do want it to stick on my window for a very long time. So that's what I'm doing So I'm going to just move this just a little bit to give myself a little bit of room in between here each one I'm going to push continue and I'm going to show you which material I like to use And for what works for me is actually I go to the pressure and I do more And you want to make sure that you have a very sharp New fine point blade in your clamp a here That is really really important You know what I'm going to do a giveaway actually a company reached out to me and sent me some blades And they sent me quite a few to do some giveaways for I'm going to do a giveaway for these blades to enter this all you got to do is leave a comment down below And then if you wouldn't mind just share my video share the heck out of it guys Tell people about I create crafts. I love doing crafts. I love showing you guys how to do things Leave a comment down below about this video if you like it that you shared whatever and I will Randomly pick one person from the comments down below to win The package of blades and I absolutely love these blades. I bought mine over a year ago And I still I probably used one or two blades. So they go a long way. So anyways, go ahead and do that I'm going to go ahead and cut this out on 651 vinyl. I'm going to put it on a green mat. I'm not going to show you guys that because probably know how to do that I'm going to cut and weed it out and then I'm going to show you the process outside of my House on my car. You have to clean it and do all that kind of stuff get it prepped So I'm going to show you all that in case you guys don't know stay tuned Leave some comments hit that like button hit the subscribe button if you're not subscribed already and let's get going guys All right guys, so we're outside. I have my transfer tape on here's my truck Do not mind my messy truck. Like I said, we're in winter right now It's actually eight degrees outside. So I have to go really quick Not a good idea to do a transfer at eight degree weather. I would do it warmer But I'm not waiting until summer. So this is where I'm going to be putting it I did not measure it, but I would highly suggest measuring it. I kind of had an idea of how big I wanted it So it turned out perfect. So I'm just going to take some windex. You can also take alcohol and Clean your spot really well And like I said, I did a video on this before With my truck with the back top being on with a really huge back off one And that one went really well Unfortunately, I got rid of it. So I need a new sticker on here So I'm just going to show you guys really quick how to do this. So you just want to make sure it's really clean And very dry and like I said, I'm using 651 permanent vinyl So this will not come off if you do want it to come off It it is kind of easier to do all you do is take like a Exacto knife and you scrape it. You don't want to scrape your window But you want to scrape the vinyl off which it does come off rather easy So I have a clean I have a dry have my transfer tape on all I want to do is take this off And I gotta hurry because my fingers are freezing As take this off So here's the design if you guys can see that And I'm just gonna I'm trying to place it. I should be inside my truck, but I have this on here. So it's hard to take it off So I want to do it high enough You know that people can see it And also try and keep it straight So I'm gonna leave it there. It's on it's not going to come off anymore So what you want to do is I forgot my scraper But ideally you want to take your scraper and you want to scrape it on really well I forgot mine in the house. You want to make sure you don't have any bubbles or anything in it? And then just peel back. Oh man, it's so cold. I can't even grab it My fingers are frozen Ideally like I said, you want to do it in warmer weather But just like that and if it does come off just use your fingers And push it down This is not any weather to do it for So just like that. I love how it turned out I might put one on the side over here too as well But I like this just think when you're putting on where you want your placement to be that you're not going too high or too low I do love how that looks. So if you guys like this tutorial, please give me a thumbs up hit that subscribe button Make sure you share my video. You will have a chance to win The blades which are really amazing. Like I said, I use maybe one or two blades that for the whole year So it's a really good opportunity If you have any questions or anything on this, please let me know go back check out my other videos I do have a couple of other ones. I show you how to make different ones and then how to sell them to actually I sell mine They do really well at the stores that I'm in actually sold out a bunch of times. I have to create some more So happy crafting everyone and give me a thumbs up. Bye


What's the word y'all? I cannot believe that Miami Heat did it man. I thought with them being down by 15 plus points in this game that this, the main topic of this video is about to be about the Knicks. If we gonna still talk about the Knicks, but I thought we gonna open up the show talking about the Knicks dominating the Cavaliers. But instead we just saw one of the biggest choke jobs ever. And absolutely I'm gonna be hyperbolic today because I'm on edge. The Warriors just won their first world game of the series. Everything is happening right now. And I'm filming this right after that game ended. So if there's post game interviews, rumors, whatever, whatever, whatever. Just know I ain't seen it. I'm just out here reacting to the games that we saw today. I picked the Milwaukee Bucks to win the championship this year. And I'm not alone. They were the Vegas odds, Vegas favorites to win the 2023 NBA championship. And they just lost in the first round. And five. And five. Now we dropped the video a few days ago talking about how injuries, of course, have played a big part in a lot of these series so far. Giannis went out early in game number one, he's game number two and game number three, came back and it has not mattered. There's so much to talk about in this game alone. But I wanna start off by giving the flowers and the love to Jimmy Butler, to Gabe Vincent, to BAM out of bio for getting this triple W, even though he fouled down the strikes and to Coach Eric Spostra. Because he coached circles around Coach Bud. He made Coach Bud look like a JV in high school coach while he's one of the greatest of all time. You know that list of the top 15 coaches of all time, Spostra's on there for a reason. And you saw that in this series and you saw that specifically in game five. Now we got the Knicks versus the Miami Heat in the second round. This is, and we gonna talk about it, we gonna preview or whatever, that's not what today's about. But I just cannot believe we just saw an AC take out a one for the first time in a very long time. Cause we don't count the year that Derrick Rose tour is ACL. So I just cannot believe everything to happen in this particular game. Another example, where Coach Budenhozer was often, simple as that, I was trying to find another way to put it as simple as that. We talked about that in game four, how the Miami Heat went on a 13-0 run in this game. And he never called a time out or nothing. And of course, the Miami Heat went on to win that game.  Exam way, using a challenge in the first quarter. I've never seen a time where a challenge in the first quarter was worth it. Number two, the Miami Heat have a sideline out of bounds. And I can't say the only option is to throw the ball at the rim cause there was a lot of different options, there was time left. But he yanked Brick Lopez out of the game so there was no rep protection on the shot that Jimmy, Jimmy probably had no business making. But there's no rep protection at all on that shot. And then lastly, these are just the three of us on top of mind. I think if we watched that game, I could be like, oh, there's a moment, oh, there's a moment, oh, there's a moment. And lastly, not calling the time out at all in overtime where his team didn't know what to do at all. Giannis is scared to get fouled, that's something we gotta talk about. He's scared to get fouled, he gives it up. And I think it was to Chris Middleton. Chris Middleton gives it up to Grayson Allen. Your season is on the line and Grayson Allen has the ball with three seconds to go. If I'm the coach, I'm running to half court if I have to, to call this time out because no way, they didn't get a shot up. Because Grayson Allen, it's not in his DNA to try to make this, he Euro stepped with half of a second left on the clock. Like there's no way Coach Bud coats his game like that, bro. Also I'm realizing that my light and it's gonna be weird, I don't care. I'm already here, we already rambling. The offense in this series in general stuck. And the half court, a lot of the times in this fourth quarter and over time, not over time but in the fourth quarter, we got Giannis pull up mid range jump shots. And it's like, yuck, obviously. It's like yuck. And if your coach's supposed to you, anybody on the Miami heat, you like, we love that. We want him to take that shot over and over and over again. This is one of the first times, at least on the top of my mind again, I'm just reacting. Where I felt like Giannis was afraid. I remember in the NBA finals last year he had a game or maybe it was the finals of conference lines. Let me fact check this, where he struggled so mightily from the line and he went up to the podium and he said, I'm still gonna do what I gotta do and get to the free throw line and let the results be the results. So I'm summing it up, but that's something I remember. And for 85% of the game, that was facts. He ended up shooting 23 free throws tonight. 23 free throws. So he 100% wasn't afraid until it got down to those clutch ones. You know, on the jump ball, he almost turned it over because he saw two people coming at him so he wouldn't have to get fouled. And luckily Chris Middleton saved the ball and ended up getting the Drew Holiday Sands and Drew Holiday missed the free throw. And then the last one is the last play of the game. Jimmy Butler kinda cut off the sideline but it wasn't enough that would naturally prevent Giannis from doing what he could do. But since he cut it off with some contact, this is just me speculating, I don't know. But it felt like Giannis thought that maybe a whistle was gonna come and he would much rather give the ball to Chris Middleton to do something instead of getting fouled to go to the free throw line. Now in this one, he shot 23 free throws and he missed 13 of them. He shot 43% from the line. He makes half more of those. A couple more of those, they win this game undoubtedly. And again, he's never been a good free throw shooter. I mean, we had the games in the finals where he dropped the 50 piece and he was parked from the line where he turns to Steph Curry or Steve Kerr or Jose Calderon from the free throw line. But that's not him. But even this is not him. Missing 13 free throws to shoot a 43%. That's what 25%, I'm not 25% points less than what he normally does. Like overall, if you were just looking at the stat line, 38, 23. Like that stat line is crazy. Like it felt like he was giving it all on the line. And for the most part he was, but there was these moments where he's afraid to get fouled that kind of, in some cases, swayed this game to the other side. Giannis is the best player in the world. I've set that for a few years now. But I am 100% certain that after this game or being one of the few people to be a one seed and lose to eight seed, people are taking, they're gonna take that title away. And I'm not gonna say they're wrong. And again, I would still say he's the best in the world. But if somebody said no, he's not the best in the world anymore based on this, I can't say they're wrong. Now again, he's dealing with the buttocks injury, the lower back injury and all of that. And that probably, not probably, that obviously played a major part in the series in general. I can't say specifically this game. But the series in general, again, he missed two total games in the series. And then I might as well put it at three because he got injured so early in game number one. And again, I don't know what the severity is, but maybe, you know what, I'm not even gonna say that because I can't live in a world of speculation. I got too many people watching these videos. But this is a historic collapse even with Giannis missing the two games. Because, I mean, they showed us all season long in the games that Giannis hasn't played and whether it be this year or last year, whatever, this is the same core that they can pretty much get it done. I'm not saying that I would pick them to win a championship if Giannis is not there, but in a series like this where it is the Miami, let's be real, the Miami, he played a great series, but the Miami, he'd are very far away from a great team. They just are. I mean, I know he fans are riding high and they should because they just pulled off something legendary. But until the playing or the win against the Bulls, the playing 98% of the he fans were done for the season. Like they watched them in 82 games and came to the realization, oh, this is a lost year. You just lost to that team in a seven game series and it didn't even come close to being a seven. It was a straight up five. I don't see how Codesbug keeps his job. I had already heard some people kind of calling for him because even basically saying that he had been carried by the postseason run that ended up in a championship. And again, being a championship coach is something that only a few people in the world can say. So it's not like, I can't say he's the worst coach of all time because he was the coach on the team that won a championship. And yes, Giannis dropped 50 and was again turned to Steve Kerr at the Frito line. I mean, Bud still coached that series and coached that team. So he's very far from being awful, but in this series, he looked close to it. Nick's versus Heat. Not me honest with you, Spoe.   So you're not about to go against a Codesbug level coach. You're going to get another one of the best in the league right now. So I'm very curious to what that's going to look like because it's going to be a chess match. But again, we're going to get to preview on that once we get closer to it. I do want to talk about this series that I would say that of course now other than the Bucks, the most disappointed team in the playoffs for me has been the Cavaliers. This is a series where I picked the Cavaliers to win in seven. I knew that the Knicks are really, really good, but I ultimately won with the Cavaliers to win a series before it started because I thought the Donovan Mitchell and his star factor would be a little bit too much. And I overestimated that and I underestimated the depth and the coaching of the New York Knicks because my pick to be in seven, another situation where it wasn't, it wasn't even close to being seven. There was no moment in this game five where I felt like, okay, the Cavs, here they go. I think there was one moment actually where a third quarter DG hit a three to cut it into whatever. And I was like, okay, maybe everyone's about to come. It did not happen. It did not happen at all. The Knicks have had this season or this series very, very easily. And we got two bad games to RJ Barrett. The last time we talked about the series, we were talking about how RJ hadn't showed up and since then he's been playing phenomenal basketball. I did not expect for Mitchell Robinson to be the best big man in the series, simple as that. Remember, Jared Allen was an all star last season and Evan Mobley just finished third in DPOY this year. So like, I was pretty confident that at least one of those dudes would be able to outplay Mitchell Robinson. Absolutely not. Mitchell Robinson was everywhere, all series long, whether we could protect in the paint or getting extra opportunities. Tonight specifically hit 11 offensive rebounds, 11 of them things. And Ham single-handedly made a front court of two seven-footers look small, look tiny. And I also didn't account for basically Jalen Brunson being the best player in the series, simple as that. Like tonight, again, he didn't shoot the ball crazily, 36% from the field, two for six from three. So it wasn't a great shooting night for him, but he outplayed Darius Garland. He outplayed Donovan Mitchell in this entire series. And this is so crazy because Julius Randall didn't have a good series. There's no reason the Cavs shouldn't have at least made this a six-game series, a seven-game series or something. Julius Randall got injured in this one. And in one of the games, he didn't even play in a fourth quarter because the other people on the roster was just playing better. And that's a testament to Tom Thibodeau being a good coach. And there's a testament to the depth because again, there was no quitting grimes for the last couple of games. Didn't matter. Q Grimes has been started for a great majority of the season. They lost them. Didn't matter. Josh Arcelin to that starting line today. Before that, we saw Miles McBride minutes the game before. Do you know how demoralizing or backbreaking it is for your team to get out rebounded on the offensive glass by 13? 13? I love what Josh Hart said to Mitchell Robson. He told Mitchell Robson, you get rebounds. I get rebounds that like break hearts. I don't know the exact quote, whatever. And it's true. When you gotta do like Josh Hart, let me see who they list Josh Hart as. A six-fold wing crashing in, getting offensive boards. That is demoralizing. And now we have Mitchell Robson getting 11 specifically tonight. It's like, boy, what are we supposed to do? I'm realizing that my lighting has been all over the place this video. Hey, you got to accept it, man. It's one o'clock in the morning. We gotta film. The only bad thing for the New York Knicks as they came out today is that the fact that the Miami Heat won their game, which means they won their series, which means that this series has started way earlier than some of the others. It was like, with Trey Young hitting that shot, Joel Abbe gets an extra couple of days of rest and you would have won at a couple extra days of rest for Julius Randall's ankle. We don't know much about that other than he didn't play the second half. And again, Julius Randall is a all NBA, he might make all NBA this season. Didn't need him this series. Is that crazy? They didn't need him this series. That's unreal. But again, we about to see some coaches and juggernauts potentially go at it.  They say a series does not start into the role team wins a game and the Warriors did that. The Warriors did that. Who expected a 20 piece from Trey Ma Green? Because if you got your hand up, I'm saying you lying. Trey Ma Green coming off the bench might be the recipe for a run. Honestly, it's so crazy to say it out loud, but he was super critical today. And you know what, I'm allowed Trey Ma to talk his stuff. You know what I'm saying? Sometimes Trey Ma be having those games where he overfolded and then he finally hit one and now he talking and talking. I'm like, bro, you just missed the four previous ones. Be quiet. But nope, today you can talk all the trash he wants because they just took the series. It's not over obviously, Kings fans, but like now you just need to win one and we're going back to Chase, a place that we don't lose that often, you know? We finally got a resemblance of a solid DeMonte Savonis game, but it didn't matter because Kavanluni having another 20, I didn't realize he got to 20 rebounds. I mean, let's be honest with you. Once this Heat Bucks game went to OT, the Warriors game kind of went on a backburner because this was more important, but I didn't realize he got up to 22 rebounds. I think the super unfortunate part for the Kings perspective is that some of their worst shooting nights of the entire season have happened in this series.  They had one of the games that they ended up winning. I think that was game two to end up winning was the worst shooting night of their season. And then game four was like the second worst shooting night. And I think this one might be in the top five or bottom five as well. So young team missing a bunch of shots. It's kind of unfortunate. I think that De'Aaron Fox was dealing with his injury and we didn't even get like a crazy Steph Curry game. We got the last possession where he just running around the court until he eventually got a layup. That's when I started to film this video because I let that swaps. But yeah, the Warriors took that game and now they're one game away. The Lakers lost their game. So we going back to the Staples Center to see that one. I'm just excited at how great basketball has been in the first round of the playoffs. I'm a little bit afraid that like because this is so good that the second round might be mid, but I'm not gonna woo it because I need it to be great. So shout out to Jimmy Butler and Miami Heat, just ridiculous history being made. And I'll see y'all tomorrow.


Asalaamu alaikum and welcome to another episode of Making a House a Home with myself, Sana Ararji and our guest, Afima Muhammad, a qualified life coach and an NLP practitioner. Asalaamu alaikum, Fahima. Now today we're going to be carrying on talking about Ramadan and discussing family structure and community and how that fits into Ramadan. Can you explain how family structure and community is important during Ramadan? Well, let's just remind ourselves first about observing fast and how it's prescribed upon us and how we have to become pious in Surah Al-Baqarah. It does actually highlight that. And Fatima Tazahra alaihi salam says that fasting was made obligatory as a means of establishing sincerity. Now when we hear these sort of hadith and Surahs and these statements, we sometimes just overlook it in a general sense. But I want to bring it back to the actual home, the actual family, the actual structure and community. And also we should know that there's Imam Sajjad states in his Rasala that and the right of fasting is that you should know it is covering, that it is set upon your tongue, your hearing, your sight, your tummy and it's protecting you from the fire. Also, it is quoted that obviously with fasting, Imam Sadaq alaihi salam has said that Allah has made fasting obligatory not just for the rich and not just for the poor, but for us to be as equals. So when we take all of these sort of hadith and examples that I've just highlighted and you bring it back to the home, I'd like to sort of like say that we receive a lot of information and instruction generally about Ramadan as to what is the significance and what we should do and what we shouldn't do and what nullifies Ramadan, which is very, very crucial. It's very vital information. But at the same time, we need to know the full human condition and the experience that we need to sort of be aware of during this month. So for example, when we have increased in our Ibadah and we increase in our reading Quran and our fasting and worship, extra prayers, charity, the heightened consciousness of Allah, then obviously we are thinking that we are doing what we need to do for Ramadan. But we also have a responsibility for our families. We have a responsibility for our homes, for society and for outside the home. And that's really important. Islam is for us, but for us to share and for us to integrate and for us to learn as well as teach for us to be influenced by the good and to influence others.  But let's just consider the challenges and the tests that are faced. For example, the reverts. Yeah, of course, we need to consider that as Muslims already, we have maybe friends or we might see someone in the mosque all the time. But how much are we actually, you know, opening ourselves to that person to that type of community? Because, you know, everyone's segregated. Unfortunately, it's not just in Ramadan, but generally our mosques, our centers, the people that we associate with, the cliques, the friendships. It's very much our own kind, our own culture. It's not Muslim as a ummah. Yeah, it's not even humanity, people, because the only way sometimes you can influence is to actually go and integrate into outside your own religion, outside your own culture for you to learn and for you to teach. Yes, of course. Yeah. So family structure and community is important and vital information and awareness that we need to have.  Yeah, of course. I mean Ramadan can be a very lonely time for converts or reverts. And, you know, not all of them were blessed to be brought up in a Muslim household where they've seen Ramadan every month. They've observed the fast as a family, as a community, and they've had that beautiful spiritual closeness. So in a way, you know, could we do more to bring them into our homes as in allow them to be involved so that they can fill part of the family because a lot of the time, you know, it's mainly them that has converted and their parents are still non-Muslims or their siblings are non-Muslims and they're lonely and they're kind of isolated. Of course, it's definitely an issue that we need to be aware of and raise ourselves. I know the mosques and the centres, they have, you know, circles and societies for reverts and they might mix between them. But they themselves want to be open and integrate into the communities around them. But they can't do that and invite themselves. So everyone around who is already Muslim, who is already, you know, in a community, need to take upon that responsibility to take them and invite them home, to give them that, you know, few days of knowing what it's like to be in a family in a Muslim community, whether or not they're with their families or not, some of them, you know, are still living, like you said, some of them are alone completely because they've actually taken themselves out of even their own families and communities. So this is something that we do not raise enough and we do not give awareness as to what is it that we can do as individuals, as families, as a community. And what more can we do? So like I said, you know, we need to pray fast, give charity.  It is so much more and that's what we need to take from this holy month.  So to invite someone to give them an experience, imagine. Yeah. And the first experiences are very important because if someone is perhaps considering becoming Muslim and they might visit a mosque and just that moment of stepping in the environment, the feeling that they get, it could create either a positive or negative. And the best time is Ramadan because it's where everyone gathers up. So the mosques do have a role and a responsibility to to all Muslims of all faiths. And I feel like that is slightly neglected in some cases. Do you feel that? I'm not sure about it being neglected. But what I can say is that when it comes to integration, I think the responsibility should be on us because we only go to the same places all the time. And that's not a learning for us in our new generations. Okay. I know we go for convenience a lot of the time because it's close to us. But then why not make it every week? We go to a different center, whether it's the Horseshoe, whether it's the Iranian, whether it's the Iraqi, whichever it may be. Yeah. Why don't we learn in that way? It's a great idea, actually. You know, do it that way. And that's when you can see more. That's when you can learn more and open up. Put yourself out there, even if it doesn't come to you. Yeah, definitely. And I think that's a better way of looking at life. And actually, because we shouldn't just be Muslims, united by culture and country, we have to be united more so as humanity, you know, looking at humans as one, as being as one. So our religion would only be the way it is because of the way it was spread in history. It didn't just stay in the Middle East. It went much further around the world. That's why it's so widely spread. But now we are even more enclosed. We're so scared of, you know, letting our children play with another sect or another culture or another background. That's how we've become in this modern day and age. If your home is solid with good foundations, good learning, good values and beliefs, you can let that child go out and they will learn openly as to what is right or wrong because that's what they've taken from the house. But to keep them caged and closed and only in the same setting, in the same situations, all the time seeing the same faces, that's not learning. That's not opening up your mind. That's not opening up anything. It's same again. It's just, you know, remaining in this bubble. Yeah, we're limiting ourselves really. It's very limiting. Yeah. So going to different mosques, you know, there are programs everywhere and they're in languages now which we all speak in English. So we're very fortunate. There's absolutely no excuse, especially in London. So, you know, take that time to experience that. And with Reverts as well, I find that we need to sort of be a little bit more, you know, sincere as to what they're going through. Because it's not just about experiencing a thaw. It's about the family unit. Yeah. But our own family unit needs to be solid. There's no point in you fasting as well. And there's no point in you praying and giving charity when you are at home grumpy and you're not even giving that attention and the time to your own family. Yeah. So those things are really important. Yeah, definitely. And then also we have others that we need to consider which are widows, single parents. That's a very testing time for them. Yeah. And this is the reality, unfortunately, in our society today. We have people that have been separated. So how do they celebrate? How do they benefit? How do they live? You know, because families, you know, are together a lot of the times during the holy month. So what is the structure there? Yeah, especially when there's children involved. Yeah, single parents, it must be incredibly difficult because it's incredibly difficult. You know, perhaps they had negative stereotypes towards them. Maybe they've been outcasted from their community and they're left on their own. And that could maybe make them go, you know, away from Islam. Absolutely. And so it's so important that we, you know, we try and stick together as a community, really, to help even if we see our neighbour who is a single parent or perhaps a friend that we know. Yes. You know, we should maybe invite them. What, you know, what bad would that do? Invite them, bring them to your house, let them see, be around you. That's true. I mean, you raised a very important point again by saying that, you know, people when they're in the situation automatically they are outcasted because of the situation they're in. A lot of the times it's not their fault. Yeah. And don't forget, we all live in glass houses and anything can happen to us at any time. So, you know, the more we lend, the more we reach out, the more we give and open our doors. Inshallah, then, you know, less bad and less anything will come to out to us. When you're like, okay, no, well, she's divorced now, we can't really have her round or he's, you know, a single person now we cannot invite them anymore. And that family is now, you know, not a family anymore. You know, we need to open our minds. Let's just talk about the reality of today. This is what we're doing. Yeah. And we call ourselves Muslims and we're fasting. Yeah. And then we have this sort of, you know, way of thinking and believing and behaving. So we need to really open up Islam and the teachings and the meanings and how we live day to day. This is the whole point of the show. Making a house a home because it's daily challenges, daily routines, daily habits that we have created, the way in which our mind is, the way in which we are being as humans to ourselves, to our families, to where we look at other people and all these stereotypes exist, all these cliches, all of these things do it because we create it. It doesn't mean if it exists, we have to conform. We can be the one that breaks those barriers. We can be the one that actually makes the difference and it only takes one person. So, you know, I'm just going to carry on like normal and I'm going to still invite that person or I'm still going to have them as a friend or I'm still going to be, they're still going to be part of the community regardless of their situation. And that is the difference that it makes to an individual. And that's when you become a leader, that's when you become an influencer, that's when you become, you know, a legend in a way, because you are creating something for generations that are watching your children and society and communities. But we don't do that, especially in our society and communities, especially amongst the Muslim Ummah, is very, very segregated, very, very much to their cliques, very much to their country and whatever language they speak in. Within their comfort zone. Within their comfort zone. It's whatever and they don't want to go out of that because it's safe and it's secure and they don't want to explore. But they're missing out by not exploring. They're missing out. And you know, just one person, it could affect them and change their whole life around and maybe their children and then their children's children, they don't, you don't realise the impact it could have on one individual. Until you do it. And I think we maybe tend to listen to what others think about us or we're scared to be judged. But ultimately we should just be scared to be judged by Allah. Absolutely. And that is our priority. And I think sometimes it's hard, many people tend to forget because they're very well known within their community and they don't want to be seen in a negative way. That is so true. That is absolutely true. We are dictated and defined by other people and the so-called community and society that surround us. But how much harm is that actually doing for ourselves, for our future generations? So Ramadan needs to be something that is reflection upon ourselves. It's the way in which we think and live. And when it comes to our family structure and communities, we cannot, we cannot be segregated anymore. You know, these sort of things that happen are so backward. This is what Islam is so much against. And if we're doing that to our own kind, what hope is there for the future? What hope? And if we want, you know, the outsiders outside our religion to look at us in a particular way, then we need to set those examples between us. We can't be brothers and sisters between us. We say those words, brothers and sisters. What does that mean when we don't live it? We don't show it to real brothers and sisters, let alone the ones outside the community. So that is something that needs to be considered. And the other group of people I would say that we need to obviously also consider are the elderly. They are very lonely because of the time of life that they're in and their stages. And they can still, you know, be invited in our mosques and our societies and communities to even give advice, to even give talks and lectures, so that they feel part of something still, that they can actually advise because the youth need that wisdom from the elderly. We can learn so much. They're so useful, but we just forget them too. And that's very lonely times. And a lot of them may not be able physically, but if there's a way of doing it, it can be done. How Ramadan used to be. And what little that we had that we used to still celebrate and appreciate and, you know, do things during this time and worship with so little. So that's the other thing that we need to consider. And when we invite the elderly, we can learn so much history. Yes, very true. And we can really like, you know, appreciate what we have. And even we think in today's society with technology and with all these advancements that we saw ahead, but actually things were done a lot more cleverly before. And community was a lot stronger before. We have a lot to learn from them. I think we think that we don't need to listen, but actually from someone who is elderly, their life experience, the knowledge that they've gained throughout their life experience is much more than ours. So we can pick the things that are good and maybe use it, incorporate in our own lives and learn from the elderly. They have a lot to give within the community. Well, you know, the spirituality and essence of Ramadan is so much more than you thinking about your food and just thinking about your tiredness and just thinking about those long days and thinking about work and children and home. Because, you know, it's about people. That's what makes us connected is people around us and inviting all types of people around us. Our homes shouldn't be with just one color. It should be all colors. It should be a rainbow. Our community centers should be that too.  And even if they're not practicing, even if they have no idea about religion, it should still be open. Definitely. So these are the things that we need to consider when it comes to family structure and communities. And we as individuals really need to take more responsibility about that. And the, you know, the, you know, speakers and the, you know, society, sorry, the community centers, they need to advise about this too and open up people's minds about this so that we are aware of it. So when we see somebody we don't have to be even friends with them.  What's your name? Where do you come from? What do you do? And there you go. There's a friendship there. And you can learn from each other. You can, you know, advise each other and you can have something different. And everyone has something to give. So don't underestimate anything. And your situation will never be the same. Just remember that.  Well, you know, thank you very much for Hima. I'm sure the viewers have learned. So inshallah we'll be going to break now and inshallah you'll take the information on board. And maybe during Ramadan we can perhaps do something different and invite someone into our homes that we wouldn't normally invite. But inshallah we'll come back to answering some questions from our dear viewers. As-salamu alaykum wa rahmatullahi wa barakatuh. As-salamu alaykum and welcome back to the second part of making a house a home. Now we've been discussing family and family structures and community. And we have a few questions from our dear viewers. Now the first question I have for you for Hima is Mustafa and he says he's a revert that's married with his own family. But he remembers when he was new to the religion and he found this time to be very lonely even though the mosques did have classes for reverts and a few friends did invite him to break his fast. However, people in general are very much used to being with their own kind. So how can we change this as I do not want to see the future generations conform in the same manner sorry. Yeah, I mean at the same time we cannot change other people's ways. So we need to basically start with ourselves and even as a revert now having your own family you can be that change and you can be the one that actually you know invite other people from all different cultures and sex and backgrounds and at the same time you know show others we need to take responsibility as Muslims generally and it's really sad that we still don't year after year decade after decade generation after generation we're still set in our own ways of when it was time in you know in Arabia where everyone was like you know in their own sort of you know type of background with regards to their sect and which you know which sort of you know background and you know culture and tribe they were from. So that is very back with thinking and I think it's time that we really need to reflect and we need to take it an actual goal and to make it an outcome that this Ramadan we're gonna set a goal that we're gonna find someone that is outside our community that is outside the norm and actually bring them home. So you know it that's if that's what it takes out what it takes and if each one of us that's listening you know makes that a priority it can be done yeah and that's when you can just automatically it'll be the norm. So it's it's you know it's us as individuals taking that responsibility upon ourselves and the enlightenment from that and the empowerment from that is just you know so dramatic and we don't realize that when we do this our children are watching and that's so important that's a learning in itself without any words being spoken sitting amongst people from all different cultures and backgrounds sharing and praying and eating together and that's what it's all about yeah not just the same faces with the same language and the same thoughts you know because even the conversations can be quite dead and boring you know this can bring out a so much more elaboration on information and knowledge learning from a person's past and their journey and that is teaching in your home and you say you don't have time well there you go invite someone that's not from yourself you know in the same way and that alone is a lesson yeah so you want to change communities you want to change generations start by doing it yourself yeah and and again I advise you know that each Muslim person takes it upon themselves to actually do the same yeah would you say maybe perhaps as a like if a revert was to go to a mosque they should perhaps try and try and push themselves to get involved like don't be scared yeah that's the other way of break through the the the barrier of fear and think you know what I'm entitled to be here I'm a Muslim yes I'm not going to allow anyone to intimidate me that's true and and and then build build your way up in that way whereas if everyone runs away because they're scared no one's going to be going that's a very good point of looking at it too because even if people are not inviting you need to be the one that you know looks like you can be invited by putting yourself out there you know speaking about this even in one of the lectures and say can I even have you know a space to talk for five minutes yeah and you know raising this awareness in one of the centers when there is you know one of the Ramadan you know series or whatever it may be in the mosque yeah so that people have that yeah definitely that could be something because a lot of people is also lack of knowledge so they probably think oh they're taken care of everything's fine they don't really realize that actually no there's a lot of people out there that are lonely and we're one of them yeah so um whether it's even getting a group together in the mosque it doesn't have to be in the homes so that you get a group of people together the regular to come and you know be part of this group in the society and then you know raise it in that way yeah so there's various ways of doing it so yeah that's a very good point inshallah yeah that that's a great advice I mean um the second uh viewer we have is uh anonymous and uh they say I'm divorced I'm a divorced woman with children and Ramadan is a very testing time for my family in my home it is nice when I invite family and friends but to receive the same in return is not usually extended because previously family have stopped others inviting me how can I deal with the situation well that's lack of knowledge that's lack of you know um sort of way of being um yes when there's a divorce when there's a separation of course there's a separation of family and friends but it doesn't mean that it goes to the extended there's a lot of families that still mix remember especially if there's children involved yeah they're still related in cousins and relatives and even as adults you still have friendships within the other side that you know you're divorced from and if people want to obstruct that and if people want to segregate that and then they want to you know cause barriers then it's on them yeah at the same time it just how many highlights that they're the ones that have the issue they have the psychological you know um sort of like imbalance because you know obviously that person's probably intimidating or that person you know just feels like you know they just want to keep that person apart for whatever reason but that again is not living and being as a Muslim because things happen yes you know there are separation and there's divorce and doesn't mean you have to come together as one but there will be scenarios where you might still mix and you know mingle for you to go out there actively and say I don't want that person in my space when I'm around then what does that say about you just in a general setting what does that say about you let alone in an Islamic home for you to be out there speaking that to somebody and you know causing that barrier you know creating a broken relationship yeah or trying to break a relationship a friendship what is that and that's one of the worst traits of a person yeah you know at the end of the day it shouldn't matter who's there and who isn't there even if you don't want to have a conversation in friendship just be civil and say salaam alaikum and be you know respectful yeah you know normally in these groups and settings they are you know they're quite a few people but you can still be in the environment where you're with somebody who is previously your family or relative and because of divorce or separation they're not your family anymore but if you want to cause more rift and more fuel to the fire then it's only going to come and affect you anyways yeah definitely and it could you know that would affect the mental state of the mother and then how she looks after her children I mean the impact it rolls over of course the main thing is that when you're stopping the adult you're stopping the child that also is your relative yeah so what is that people don't see it in that way punishing the child you're punishing the child because of your own ego yeah whatever it may be or your own issues whatever so that person needs to sort themselves out but you as a single person you keep keep inviting you know your doors are going to be open and Allah's going to open more doors for you in fact your life's going to be elevated even more so because people who wants to put you down or you know press you in a push you in a certain wall or keep you closed you're going to be opened up even more yeah and you're going to be in their faces even more whether they like it or not I know yeah that's what happens would you say it's better not to retaliate for example if someone is being like that you know us being human beings we try to you know question question and we might be can't confront them or we might think you know what if they're doing that to us I'm going to do that similar thing to them we should try and let them be let them be and just think you know what we're doing this for the sake of Allah we shouldn't care about if they want to be negative they want to be like I will pity that person who feels like that because only hurt people hurt people and sad people you know are like that to other people and they obviously have their own issues so you pray for them you know that's the only way to rise above other people is just always show goodness no matter what yeah positive comes out of it no matter what yeah and that's where your strength is that's where your protection is and only prayer will be able to give you that strength to do that yes prayer is your power definitely so that's what you need to remember and other people that want to put you down and you know seclude you and you know make your life even more worse than what it is that's on them they're being judged yeah you know you cannot control that but you don't need to retaliate you don't need to counter or confront if the occasion is there and you feel you can speak about it depending on the relationship you can address it in a very civil way yeah and a very respectful way maybe just to educate them but most of the time these people are doing it out of spite yeah out of their own ignorance and they know anyways and be the better person I guess better person exactly okay that's great and we have another question from Miriam and she says in our culture and societies we tend to not mix as much so how can we even integrate outside and teach others about Ramadan and its benefits let alone that our religion represents unity amongst many other wonderful things yes again we need to take responsibility ourselves it's no point even looking and pointing fingers at anyone else and say they don't integrate you yourself integrate yourself put yourself into centers that you wouldn't normally go to go to mosques that are outside your community and culture and that alone will get you being seen and hurt and then invite them to yours yeah do it in that way and you know like I mentioned before you know trying out different mosques in different areas in a different week yeah you know that's a really great learning it's an amazing learning and like I said you know you make friends that way and then you call them to yours and that's where you open up and you know inshallah one day we have a group of people that will just have a center called a center regardless of its name for where it comes from and which you know culture it you know belongs to which you know background and which you know country or wherever it may be yeah it's just there for everyone yeah and inshallah that's something that we should aspire to inshallah yeah inshallah and what about the idea of parents maybe getting involved with schools and informing them about Ramadan so perhaps taking in food and say and speaking to the class and saying you know it's for us Muslims it's the month of fasting and and and that will give others that are not Muslims or maybe Muslims within different sects and inside would you think that could be something that's a very good idea I know a lot of people actually have spoken to me about this where their children will take in like a project yeah with regards to you know add it to their homework and you know discuss Ramadan yeah and you know it's really you know at you know whatever age they are when their children it's really inspiring yeah and they become proud yeah and you know it's it's a learning and it's education for themselves and others and when you're proud of what you're doing and you go out there presenting it I think it's such an inspiring way of you know you know you know bringing out this blessed month yeah and that's real integration that's real way of you know making our religion you know open wider yeah exactly and others will become more tolerant they'll understand because you're not you know you have friends from all different religions and beliefs and we're we're taught as Muslims to tolerate each other no matter where you're from so if that's put from a young age into a child's mind you know they they'll get used to that I mean there are people from the Jewish community who go into schools and they talk about their Jewish celebrations to the class so they take it out and they take their time and they go into schools and they do that and you know children will especially my child she'll come back and say we've learned about Hanukkah or we've learned about this so we as parents sometimes need to step up and not expect everything to be done for us and think you know what we have a role each of us have has an individual role as a Muslim we have a role and we definitely have responsibility yeah and we are accountable for it and we're not just going to be questioned about the obligatory we're going to be questioned about what's in between yeah and that's what's going to really make you the person and you know create you into that you know that being that needs to be something special we want to be legends we want to be you know you know creating a legacy this is the way you do it now just through money and fame and fortune or the best job or the best profession it's you influencing and being part of society in a wider sense yes definitely and we want to constantly progress yeah we don't want to stand still we want Islam we want the teaching of Ehli al-Bayt to constantly keep going going I'm not coming to an end but thank you very much for that insight I mean we've got one more viewer and she is saying Zina sorry our children do find it difficult to fast especially when they attend school and they are the minority fasting so how can we make this easier and appealing for them well as we discussed in the previous question obviously you know it's like inviting it into the schools yeah educating others because a lot of the time people also judge because of ignorance so we need to sort of you know eradicate that and you know in sort of ignorance you know sort of do the opposite and you know give the education and when our children are educating others they themselves are being educated so you say go and speak to the schools about it and yes physically it is demanding on a child again you need to you know have a process where they are fasting and you need to create individual beings so that even though they're the only ones in their classroom that's fasting you need to be there to support them as to how they can maneuver around that situation yes you know it doesn't mean that you have to be in Islamic school or you're going to have to have loads of Muslims there because in this country in this life wherever you're going to be you're going to be put in situations where you're going to be the only one standing out and you're going to be have to be okay with that and take it upon yourself to be a leader for it and let the others want to question you and follow you and you don't need to be you know afraid of it yeah so that's a different outlook and that's a different way of looking at it so i am doing this and this is why and this is my benefit you know i know a lot of people they've told me that even some of their non-muslim friends want to fast with them because it's something exciting yeah something amazing yeah how we can influence especially with children exactly it's very true i mean we've now reached the end of the show thank you so much for you welcome and for all your insight and information i again have learned a lot and i'm sure the viewers have too and i hope you've benefited from this episode and inshallah we'll be back next week with another episode of making a house a home


Unit 3 of Sailor.org's professional writing course. Now that you've completed your draft, it's time to edit and polish so that you can command your audience's attention. The goal of the proofreading process is to eliminate distractions, errors, and other red flags. But as you may be all too aware, it's not easy to edit something you wrote yourself. You read what you thought you wrote rather than what's really on the page. Now ideally, you can borrow a friend or a colleague to check your work, but if you must go alone, you'll have to trick yourself into seeing your document with fresh eyes. In this final unit, then, you'll learn some techniques to do just that. One of the simplest being to set your work aside for a few hours or a few days, come back to it later. We'll give you an editorial checklist too to help you police your writing. And you can use it in a few ways. But I'd suggest moving item by item, scanning each document for just that one issue every time. And this way, you're reading with the eyes of a critical editor rather than the eyes of the author. And hold on to this list. It'll serve you in writing and editing in the future. I hope you've enjoyed this professional writing course so far. When you've finished, I encourage you to come back whenever you like to review techniques, hone your skills, and become an ever more effective writer. Enjoy the last unit and good luck.