So, continuing with the classification of one-dimensional manifolds, let me restate our earlier theorem that we wanted to prove. So, let X be a connected one-dimensional half star and second countable that is a under assumption abstract topological manifold. Then X is homomorphic to one of the following, the open interval zero one, half closed interval zero one, closed interval zero one or the circle S1. Recall that we also reduce the k, reduce this proving this theorem to the case of proving when boundary of X is empty. Then we have to prove that is either it is open interval zero one or the circle this is what we want to prove. So, we had already reduced that case to that one. So, where do we start off? We start off with the definition and of course, later on we will say look at this is connected and it is half star which is second countable etcetera we keep on using those. So, the definition gives you immediately the following lemma namely take more generally, take any n manifold not necessarily one-dimensional manifold. Then X is the quotient of a countable disjoint union of copies of open disk of the n. It is a quotient space of some disjoint union of open disk. All manifolds are like that. By second countability argument, there is a countable atlas every manifold has a atlas. Here we have a countable atlas. So, let us call it ui fi i belonging to n this i belonging to n for X where fi i from ui to dn are homeomorphisms. Now, you take m to be disjoint union of dn as many as you know countably many whatever it is finite than finite many of this family disjoint union of dn i. What is dn i? Each dn i is a copy of dn open unit disk but disjoint union abstractly. From there I can define a quotient map now namely q from m to X be the map given by on each copy dn i take it as the inverse image of fi i. Remember what are fi i? Fi i's are from maps from ui to dn. So, fi i inverse will be from dn to X going into ui. So, they are all maps into X but they go into ui. So, q restricted dn i is fi i inverse since dn i's are disjoint copies. So, this definition makes it completely defined on q and it will be continuous because this is an atlas that means union of ui is in the whole of X this q will be surjective. So, it is a quotient as a as a fact theoretically it is a quotient map but it is also an open mapping on each of these open subsets dn i's it is a homomorphism. Therefore, it is an open mapping any surjective open mapping is a quotient map. So, it is more than that it is very stronger than it is a open mapping. So, the theorem is over this lemma is over we have displayed it as a countable union of this one. Of course, when I say I at the most I it could be finite also here. So, that is also allowed. So, next we would like to make some bargaining here. We do not want all those you know some of them may be useless and so on what are the meaning of useless. So, this is what we want to do by deleting those open sets which are already covered by the union of previous members. See we have already listed we have already made a listing u1, u2, u3 and so on. If u2 is already covered by u1 what is the point you just deleted u2 there is no need for u2 go to u3 go to u4. Suppose u1, u2, u3, u4 cover u5 then what is the use of u5 delete that one and so on. So, delete those members which are already covered by all the previous members union of the previous members this is what I want to do. So, in the listing ui, fi starting with n0 equal to w0 equal to un0 u1 that is u1 this is our notation u1 is un0 and that I call it as w0. Let n1 be the least number of for which un1 is not contained inside w0. Keep going maybe u2 is not contained then u2 will be called as un2. There is some sequence I am going to a2 will be n2. If 2 is already there u2 is already there then I will go to u3 like this. The first one the smallest one it is contained not containing so that will be n1 un1. So, call here just n1 be that one. So, you call w2 equal to un0 so that is second one w2. So, like this you put wk plus 1 define it as wk union un plus 1k. What is the property of un plus 1k? It is not contained inside wk because wk is the union of the previous members. So, I think this is very clear. Of course, if the list is finite you will stop at some place otherwise it can go go on you do not know. Even if the original list is infinite you may end up with a finite list like if the compact space you may be somebody may be a might have union infinite covering.  So, take the modified covering with the property that the nth open set coming is not contained the union of the first n minus 1 of them that is the property I want that is all. So, some simplification. In what follows we shall assume that ui phi i is a countable and minimal at last in the previous sense. So, all the extra extra n s ui s have been deleted that is it. Now, take phi i equal to psi i equal to phi i inverse from minus 1 plus 1 to x. Now, it is a special case namely n equal to 1 now we are into the n equal to 1 case now.  If you may think of x as a quotient of a distant union of countably many copies of minus 1 plus 1 that is the lemma and q is the quotient map d 1 i to x where q restricted to d 1 i is psi i the inverse of the these coordinate charts. So, they are called actually parameterizations. Therefore, to understand x what our manifold is we need to unravel what kind of identifications are taking place under this quotient map. Any quotient map is same thing as introducing equivalence relation. So, what are these equivalence classes there which points are identified with where that is what we have to understand all right. Therefore, we start. So, this is the picture that I have this is some one-dimensional manifold possibly these are all intervals of minus 1 to plus 1 and there are these parameterizations this part will cover something that part will cover something and so on. So, it is quotient of distant union of minus 1 plus 1.  So, u i j more generally equal to psi i inverse of psi j of d n 1 take a take an interval go to its image through one of the j's whatever psi j and then take psi i inverse of that. So, psi i inverse of that may not be the whole interval because that may be covering something else right. So, but it will be a subset of when you take psi i inverse it will be subset of u i. What is u i or d i I mean d 1 i right. So, it is a subset of d 1 i I have been called that d 1 i is u i also more generally okay. It is coming from u j. So, therefore, I am calling u i j this is notation psi i inverse of psi j of d 1 j okay. So, each of them is an open subset of d 1 i the first one it is subset of d 1 i. To begin with we need to know what are the possibilities for each component u i j components of u i j inside this u i is minus 1 plus 1 u i or d 1 i is minus 1 plus 1 okay. I would prefer to write it as u i and remember that they are all minus 1 plus 1. Remember u i j is some open subset of this minus 1 plus 1 okay. What are these components and how the maps are related how what kind of maps we do we get this what we have to understand okay that it is possible to understand these things is surprising because it is one dimension it is possible okay. So, let me illustrate this with an example okay. This example will tell you most of the story actually what is happening. Suppose that class consists of just two charts only concentrate on one u i and u j and psi i there and psi j there just concentrate on that part what is happening to begin with okay. So, we can assume that m itself is d 1 1 union d 1 2 instead of i and j and so on.  So, here I have 1 by 2 to 1 that is minus 1 to minus 1 by 2 okay. Now if I go from here psi i and then psi 2 inverse psi 1 and psi 2 inverse I will get from here to here from this interval to that interval a homeomorphism composite of two homeomorphisms to homeomorphism. So, what could be let us let us say that it is something like t going to t plus t plus 3 by 2 I have to be careful if I add 3 by 2 half will go to so from here minus 1 will go to half and minus 1 by 2 will go to 1 okay. So, from psi 2 to psi 1 inverse psi 1. So, from u 2 1 to u 1 2 this is the homeomorphism t going to t plus I am just assuming I am not proving anything one of the homeomorphisms. So, take the easiest one or it may be the other way around namely what is this one just minus t this one going to minus t even simpler okay. So, what happens under these two cases let us understand okay in the first case we can take alpha from m m is what m is disjoint union of these two this is notation d 1 and d 2 like this okay. So, from m I will define minus 1 to 5 by 2 by restricting on the first copy of minus 1 plus 1 it is just t on the second copy I will just shift it by 3 by 2 like this one and take t plus 3 by 2 okay. So, that this alpha will agree with these identifications okay. So, that is the whole this t goes t, t goes 3 by 2 and then goes t. So, that is why you take alpha t here and on the this part you take t plus 3 by 2. It is easily checked that this map which is defined on the disjoint union factors down to x x is a quotient space remember okay to alpha hat from minus 1 to 5 by 2. You can see that this is a surjective and it is a 1-1 mapping okay it is it is open it is actually homomorphism on each part can be verified. So, this will be homomorphism okay. So, here is the picture for that. So, this is 1 by 2 to 1 this is your u and 2 here and this is u to 1 and this is your x now I do not know what x is but I have actually got it this way okay the quotient space has to be this gets identified with this one by shifting the t going to 3 by 2 this goes to t here but this goes to 3 by 2 times this one. So, they agree a point here is identified the point here but both sides will agree on that part. So, therefore these two patch up together to give you a map from x to this whole interval from minus 1 to what 7 by 2 or whatever 5 by 2. So, in this case what we have proved is the manifold x is again an interval open interval its length may be larger do not worry we are not interested in much in the length okay. Let us see what happens in the second case which is this looks like simpler namely t going to minus t t going to minus t you are just reverse in the direction. So, in the picture I can just write this way. So, this is minus 1 to plus 1 this is minus 1 to plus 1 the other way around okay here right this minus 1 to plus 1 the other way around all right. Now a point here see 1 by 2 to this one goes to actually 1 by 2 and 1 by 2 1 by 2 and minus 1 by 2 would be the same thing 1 by 2 goes to minus 1 by 2 and that is going here. So, this one goes to minus 1 to minus t and that goes to here that is why I have written like this okay drawing picture is this way is here okay. So, what happens is this portion they are identified but this portion and this portion are different. So, you have to draw pictures like this what happens to the minus 1 by 2 and 1 by 2 they are not identified only open interval are identified this point this black point here it is not a part of the manifold because I have taken open intervals remember that. So, on this side I have not shown them at all but here I have shown it because it is ignored but here I have put a landmark here what happens these points are not identified the open interval is identified what happens to those two points they will be in the closure of both both this both of them will be closure in the this open interval part here. Therefore, half-sterveness of x is violated here conclusion is that we do not want to accept that half-sterveness is not violated but we want to say that our own assumption is wrong namely we cannot have the second case at all the psi 1 inverse of psi 2 will not be minus 3 this is acceptable to gives you a long line longer interval. So, the identifications if you have an interval like this an interval like this identification like this is allowed identification like this this is not allowed that is the meaning of if you do like that there will be a point here and a point here which will come close enough but remain distinct that means what an open part open interval has both of them is in the closure. So, there will be no interval around them which are disjoint which means that half-sterveness is violated. So, that is the gist of this one example I will tell you one more example which is much more interesting one. So, this is done so b so here I am assuming u12 is minus 1, 2 minus 1, 3 and 1, 3 to 1. So, u12 has consists of two connected components in the case a it was connected here it is not connected but it has two components u12 and u21 is also minus 1 to 1, 3 and 1, 3 to 1 this is just an assumption this is a simpler case okay. Here suppose psi 1 inverse of psi 2 is again I have to make a case t plus 4 by 3 on the first component mainly minus 1 to minus 1 less than t less than 1 third minus 1 third t plus 4 3 and on the other part it is t minus 4 third what does it mean see you have two things here one interval I did not make a picture of this one because this was much much easier actually. So, I have I will let you make a picture for yourself. So, if using this picture itself for example what is happening is there are another interval here this interval is shifted to this interval another interval here and this interval is shifted to this interval okay it is just a shifting homeomorphism okay that is the case I am I am looking at okay in which case you define beta from the disjoint union of these two copies of minus 1 plus 1 by these formulas namely take t going to e power pi i 3 t by 4 see minus 1 to plus 1 if you take pi i t it will cover the entire circle exactly once right but here I am covering a only three fourth of it so 3 t by 4 okay the first one covers that much of some other part is left out beta is the second part is covering it d d 2 is covering it namely 3 t plus 4 by 4 which means rotate it by pi rotate it by through an angle pi whatever you whatever you have taken so those two will cover the whole thing okay. So this beta is defined from m to s 1 okay there are these two are disjoint copies you can define whichever you like so the beta is a subjective mapping all right what are the points which you go to the same point they will be precisely given by this so these are the relations here therefore what happens is that beta factors down from m through the quotient map q from x to s 1 and that map beta hat from x to s 1 will be homeomorphism okay so what you have done is taking two arcs like this and then forming this kind of operation identification portion here overlapping portion here overlapping that will give you a circle so that is what is happening here okay so I think these two examples explain the whole thing whatever I wanted to tell you the the the key here is though these are look like very special examples they will take care of most of the cases so I have to make a general statement here okay there is a general fact indicated by these examples we formalize this in the following two lemmas the proof will be completely obvious however I implore you to write down it on your own so take it as an assignment okay so this is a statement let a less than b less than c d less than e less than f be real numbers now I am not even using that they are minus 1 plus 1 and all that okay so this is the whole idea because for any open interval is homeomorphism to any other open interval so that is why I am writing arbitrary a less than b less than c d less than e be real number m be the disjoint union of the two intervals a to c and d to f okay let alpha from b c to d e the b c is one end of the first interval and d e is other end lower first interval in this this part initial part of the second interval okay d e so alpha from b c be an order preserving homeomorphism like a shifting adding in the other case which which we violated what was it it was order represent t going to minus t that was not a good one so you have to take order preserving homeomorphism then the identification space namely m is the disjoint union you identify t with alpha t where t is inside b c open interval b c this you call it as m alpha why alpha because this alpha is the identification map so it is just even m alpha this will be homeomorphism to the open interval 0 1 okay now I can't write it is 3 by 4 4 to 1 by 4 1 to 1 5 and so on okay I don't care it either therefore I am writing it as some open interval namely it is homeomorphism to normally 0 1 so this is the lemma okay similarly you can have the second one which will give you a circle okay exactly similarly so this this this is the first case this is a 1 a 2 was a case which is not occurring so b case b is discovered by this one okay in the example given now I am taking minus 1 plus 1 itself to be more accurate given minus 1 less than a less than b less than 1 minus 1 less than c less than d less than 1 let alpha from this end b to 1 you see b 1 to this end minus 1 plus 1 okay the end of the first one with the initial part of the second one or the other beta is minus 1 to a to d1 so this end and that end both of them be order preserving homeomorphism again let x be the quotient of the disjoint union of the two copies of minus 1 plus 1 by the identification t is identified with alpha t on this part where t is between b and 1 and here it is using beta t is identified with beta t on the second part minus 1 to a then what you get is s1 homeomorphism to s1 the quotient space is homeomorphism to s1 okay proof is you have to correctly copy this one without using these 4 by 3 etc that is the challenge okay try so we are going to use this one so further things we will take next time so let these two examples sink properly in you next time we will discuss further proof of the classification thank you


10 Tips for Teaching Grammar to EFL Students Abroad. Teaching English grammar is an important part of any EFL classroom. Depending on the level of your students and the degree of difficulty of the topics, the teacher may encounter several problems in the classroom. However the following 10 tips should help you to prepare effective grammar lessons for your EFL students abroad. 1. Show your students why grammar matters. 2. Give quizzes that grade themselves. 3. Use games to teach grammar. 4. Put students in the role of the teacher. 5. Take requests. 6. Have students set up learning goals. 7. Encourage and reward error finding. 8. Write original sentences. 9. Use videos wisely. 10. Find it in the text you're reading. Once you have completed your TEFL certification course and have found a job that suits your plans, it is time to start thinking about the content of your future lessons. Grammar is an integral part of any EFL classroom, and these tips for teaching grammar to EFL students abroad, will certainly give you an edge over other new teachers. Thanks so much for watching. We are ITTT, the leading provider for TEFL and TESOL training courses. If you like this video, please subscribe by clicking the button down here and click on any of the videos here on the left for more interesting teaching tips for getting certified to teach English abroad and online.


This paper proposes a novel approach to improving the accuracy of EEG-based motor imagery classification. It leverages existing labeled data from multiple subjects to improve the performance of MI classification on a single subject. This is achieved through the use of a Wasserstein distance-based domain adaptation network, which measures the distance between source and target domains and aligns them using an adversarial learning strategy. The proposed framework was tested on two open, source datasets, BCI competition for datasets 2A and 2B, and outperformed other state-of-the-art algorithms. This article was authored by Qing Shanxi, Tai Chen, Feng Feng, and others.


This music was life itself very complicated, but simple It was a group of people making up something that sounded like noise and good at the same time It was infectious organized chaos On they go down the street even now singing shouting and crying through horns to an irresistible drum beat Full of contradiction and compromise. Well, it was a lot to comprehend So these local musicians made it very easy to understand. You know what I'm talking about They played folk melodies and popular songs that everyone knew Church hymns that use fundamental harmonies Singable melodies of Celtic folk songs filtered through the American Negro spirituals then that went all the way back to ancient China It was the contrapomal spires of the Baroque as danced in France and in Spain And it was African African physicality Spatial intelligence timbral colorations rhythmic discipline call in response humor and rhyming logic You get the gist Truth be told It was military marches and what and and it was waltzes and the syncopated language of ragtime And it was no less street level sexual ditties silly advertising tunes And even more these songs carried meaning connected to people's lives and the specific sound from each person's horn and Every thorny rhythm told a unique personal story That touched your soul if you let it Well, those drums spoke in underground codes that even the plantation masters knew had to be silenced those drums could talk Slaves were forbidden to play drums every place in America but New Orleans and This decision will come back to Harlem Because here we are there they were Just three decades after the abolition of slavery Playing a music in that same New Orleans led by drums and Created by working people cigar makers and tailors barbers and carpenters even tanners Who played wherever people would gather? Outdoor parades and family picnics dance and music halls on advertising carts at prize fights in mansions schools and low-class dives And let me tell you something They played the way they worked hard, but fair Direct and Uncentimental in expression This was an art an approach and a way of being It afforded any musician in the ensemble the opportunity to play whatever they knew it could make sound good And because it was down home wild Sophisticated and spoken in a language most anyone could understand if you were given to understanding things This new music was at home everywhere


Let's go on. Okay, now we go to China and probably we have to move a little bit more quickly. A lot of you've been there, okay? Now, this is, these are ancient Chinese mosques. You know, they are beautiful.  You pray there, you say, I don't believe. And you go through gardens to get there, which are zen gardens to empty your heart. So your anxieties are gone. When you go to the mosque, you don't think about anything but your Creator. And this is very Chinese. And they did it also by the explicit permission of the Emperor. Because Islam, when it came into China, which is in the days of the right regard of Calais, probably, some say the prophets or something, they were an official Chinese religion from the beginning. And we brought, and the Emperor asked for armies. He said, send me your Muslim armies. And I want them to defend me against the Mongols, against the Tibetans. And I will honor them. And he did. And so we stood by the Chinese Emperor. Okay, Muslims today, it's like, what? But that's what we did.  So what's he got in mind?  And then we'll do it again and again. So after a while, you basically become Chinese. It could still be Muslims. But I don't want you to bring your Persian or Arab women. Come and I'll marry you. You can bring soldiers that are not married. Okay, that's what he did. Chinese are smart. They're very smart. Let's keep going. Okay, that's an internet. Again, this is what we call North Eastern China. You've got different cultural zones in China. North Eastern, Northwestern. They're different. And then Southern. They're very different, but they're all beautiful. Okay, and that's their mineral because Chinese don't like tall buildings. So they don't want you to have a big, big mineral that's in the Northeast. Keep going. Okay, keep going. Keep going. Keep going. Keep going. Okay, stop, stop, stop. Go back. Please. Okay, so Chinese, you know, it's difficult for the Chinese to say some things we say. You know, like Islam, they say Yixilan. Yixilan. Okay, so and then you say Yixilan and of course you write it like fragrant orchids or something and they read it Yixilan. So that's nice, not Donald Duck. Okay, but you know, they use these different ideograms and they read Yixilan. But that means to the Chinese, you're not Chinese because I don't understand what Yixilan is. So is that good? And you have to think about this, brother and sisters, because Islam terrifies most of the white people out there, except for the really cool ones, right?  Oh my God. You know, they're going to have a heart attack. Sisters wearing their terraces. Look at all of them. You know, that's what they are, aren't they? A lot of them are that way and you've experienced that. I know my wife has a lot of others have. Okay, so maybe we also could use Islam because we love it, but we could also find another way to refer to it, you know, and that's what the Chinese do. So they said it is Islam, if you could pronounce it fantastic, but it's also the religion of which is pure and which is real. The religion of the pure and the real.  Like that sounds a lot like ancient Chinese religion. Sounds a lot like Confucius and Lao Tzu. Really, tell me more. See, that's smart. And they did that in all kinds of things. So for example, you say the unseen for Raib. You can't say that in the Chinese. So they say the color world and the no color world. Because that's what Chinese understand. The unseen means there's no colors. They're kind of smart, actually. Chinese are kind of smart. Okay, so let's go. So that's it. Then now we've come down into the south. This mosque is incredible. This mosque is in Shanghai. And this mosque is now you're in the south. See the colors are different. The woodwork is different. It's red. All of it is by order of the emperor. And you want to know how you can tell that? Go to the next slide. You see that. Well, you can't see it very well.  That's actually a dragon. Again, if you go there, you can see that clearly. It's a dragon. Why are you putting a dragon on the mosque? Why do you think? Because that means this belongs to the emperor of China. You steal anything from this mosque. You break any window. Oh my God.  Because he basically got one punishment.  See, so like you trying to do anything in this mosque, you've got to go for the black dragon and that is the emperor. So it's guarded by the black dragon. Let's keep going. This is the same mosque. You see how they, and you know this, the Chinese Arabic calligraphy, which they developed over 1400 years.  Bismillahirrahmanirrahim. Keep going. See? And then you see, they developed this incredible Chinese Arabic calligraphy. But, and you've probably seen the calligraphers to them because they come to visit us sometimes. Right? And they do it using two strokes. Right? And then they put Chinese calligraphy, classical Chinese calligraphy that tells you a story about what this means.  Islam and the cultural heritage. Because the Chinese love calligraphy and soon to believe. See, that's beautiful. And now you're speaking to them. You want them also to be able to read and up into China and take in Chinese guides who are non-Muslims. They've never been in a mosque and they're just reading everything. Primordial religion from the foundation of heaven. It's four elements. Primordial religion from the foundation of heaven. Wow. Good job. Isn't it? Isn't it? It is. Let's keep going. Okay. What is those? Looks like Chinese, doesn't it? But it's Arabic upside down. And it's the Islam Christian. Is that allowed? Yeah, really. Don't worry about it. But see, it's the Chinese like that.


Hello guys, another product review, cmpsportswear.com. Going over our Supreme bag today and one thing I like about the Supreme bag is it's big enough to take on that travel trip for two three day travel trip to a competition, but it's not too big to take to the venue as you're moving your gear back and forth. Two big giant pockets on either side to fit shoes, bigger water bottles, anything like that. The lid fully customizable, we can spot sub this lid right here, reinforce zippers all the way around. Injection molded straps, we spared no expense in making sure we had high quality straps that are going to last for a long time, so as much as you can pack in this bag they're going to take top handle for picking it up, moving it around, or if you just want to carry it down by your side and one thing that I really like aspect about this is is we got a padded laptop or electronic sleeve in here, so it's a 15 inch MacBook Pro that fits in there with plenty of room to spare and you can add some more gear on top of it. Got this nice little velcro strip on the front, whether you got like velcro patches or things like that, you give out to your team for accomplishments or training or this that and the other, and then a big front pocket that sinks down in there pretty big, so depending on how much you have in the main body versus the front pocket, you got a lot of versatility there to put quick access stuff in the front, whether you're traveling or whether you're going to a competition venue, but 22 inches tall, about 12 inches wide and about 10 inches deep, so like I said, you can fit a good chunk of stuff in here for a two three-day trip and it's still a decent sizing up bag where you can travel back and forth to the venue or competition area, so definitely check it out, cmpsportswear.com where you just square it away.


Russia is left without weapons, Moscow asks other countries to return arms sold to them. Russia is trying to get back some of the weapons it exported to countries such as Pakistan, Egypt, Belarus and Brazil. The Wall Street Journal reported this with reference to sources. Three sources told the publication that last April, a delegation of Russian officials who visited Cairo asked Egyptian President Abdel Fattah el-Sisi to return more than 100 engines from Russian helicopters that Moscow needed for its war against Ukraine. Sisi has agreed and deliveries of about 150 engines are likely to begin next month. Sources say. Those talks were part of a larger Russian push to seek help from its longtime arms customers who for decades bought Russian aircraft, missiles and air defense systems, making Moscow the world's second largest arms exporter. Throughout the year, Russia held talks with officials from Pakistan, Belarus and Brazil to try to acquire engines for the Russian attack and transport helicopters its forces lost to Ukrainian defenses early in the war. Two sources, including a former Russian intelligence officer, told the Wall Street Journal about this. The source reports that Russia is secretly trying to buy back what it sold to its customers earlier. In addition, Russia has sacrificed part of its lucrative arms export business because of the war. Particular weapons intended for India and Armenia are now being sent to the contact zone in Ukraine as the sources note. Much of Moscow's efforts to buy back Russian arms have come as the Kremlin pushed back against an offensive by Ukraine's forces in the east and south of the country. With that offensive now slowing, Russia is seeking to retake the initiative on the battlefield, although it is unclear whether the new supplies will give Moscow the resources it needs to step up its attacks. The publication added.