Hello, welcome to yet another session of our NPTEL on non-linear and adaptive control. I am Srikant Sukumar from Systems and Control IIT Bombay. So we have in our background a usual motivating image which is essentially a rover on Mars and these sort of vehicles are driven autonomously because it is not usually possible to have sort of humans controlling the equipment at all times on these extraterrestrial locations. So we want to be able to of course analyze and learn to develop algorithms that are going to drive systems such as these. So until last time what we were doing was essentially discussing the notion of positive definiteness. So this is what we were doing until last time. We were talking about positive definite functions. So this was one of the building blocks in the Lyapunov theorems. So we talked about what is the positive definite functions and after that we sort of wanted to look at a few easier tests for positive definiteness. So we saw the definition but then it's not always easy to apply the definition and so we wanted to look at some additional tests and we looked at these easier sort of tests that can be applied and we sort of also saw the connection between positive definite matrices and positive definite functions through an example and so through this example and of course we saw another example for positive definite function also. So we sort of go ahead in this vein of discussion and the next sort of function is a radially unbounded function. So let's sort of start our lecture 4.2 here. So we are starting to talk about the radially unbounded functions. So radially unbounded functions are sort of the next level of functions. As you can see you have a scalar value continuous function again. Of course this should be R. I mean I guess we use R here too. Let me see yeah we were using R here so I guess we can continue to use R here although R plus would suffice in all these cases. We don't necessarily have to use R itself but R plus is good enough. So non-negative reals for time is usually what we have. So we are of course looking at a function of time and a space variable which maps to real numbers. This has always been the setup for these functions that we denote as V which are essentially going to be our Lyapunov functions subsequently. So what do we require? We require that the function be zero at zero. So at zero value of state the function has to take the zero value. So this was the same condition for positive definiteness also. The big difference happens here. We want the existence of a class kr function which this function VTX dominates. So in the case of positive definiteness this was a class k function. So what's the difference between a class k and a class kr function? We have in fact seen several examples. So if you have a class k function it can be and we are allowed to have something like this function which sort of grows, grows, grows but stays bounded below this. So it's always increasing but stays bounded below this. On the other hand a class k infinity function has to be something like this which keeps drawing n goes to infinity. Yeah, as t goes to infinity. So here of course here you have the x and you have sort of phi x here. And since of course these functions take as arguments non-negative reals obviously you use a norm. This is what we have been doing. So just to compare again with the definition of positive definiteness it was the same sort of expression phi norm of x. The only difference being that here it was a class k function and now in the case of a radial unboundedness we require V to dominate a class kr function. So this naturally sort of implies that this domination has to be for all time. However notice that just like before we do not require if you think about it we don't require the V to be a continuously increasing function. Like a class k function is of course an always increasing function always strictly increasing function. But all we want for V is to dominate this class k function. And notice that again unlike the definition for positive definiteness here this lower bounding has to happen for all x and rn. In case of positive definiteness it was sufficient if it happened for x in some ball. Therefore we had this nice picture where beyond this ball the V was allowed to drop below the class k function. So essentially the idea is that it can give us local stability properties. So the idea of positive definiteness being defined in this manner is that it helps us conclude local stability properties. On the other hand radial unbounded is typically used to conclude global stability properties. And so we require this function V to dominate this class k function which is this guy in this particular case here for all time and for all values of the state. So this is much more stricter condition as you can imagine. However this still does not require that my function V be increasing or anything. It can be oscillating like this as long as it is bounded above its the second as long as it is bounded above your class k r function you are fine. Okay so this is sort of an example right this picture sort of gives you an example. The only thing is there is no crossing happening anymore. It always has to lie above this class k r function curve right and in such cases V is said to be radially unbounded. And just as we already stated before radial unboundedness is related to notions of global stability. Okay just like positive definiteness is connected to local stability radial unboundedness is connected to global stability. Okay what would be examples very simple examples I mean say I have x in r2 and denoted as x1 comma x2. So V x1 x2 equals to x1 square plus x2 square by 2 is in fact a radially unbounded function. Yeah why is it radially unbounded? Because if you look at this function itself and this in itself is a class k r function right this in itself is a class k r function why because it's 0 at 0 right. If you plug if you plug 0 value of this state that is x1 and x2 are both 0 which is what it means for x to be 0 right I get a 0 here and you know this is strictly increasing and going to infinity as any x goes to infinity. So in any direction if you go to infinity this is going to go to infinity all right. So therefore this is itself a class k r function therefore V is in fact a radial unbounded function also every class k r function will also be a radially unbounded function. Of course I have I can have you know a slightly more involved examples like you know I can take something like V with T also x1 x2 as T by 2 x1 square plus x2 square and here of course T is greater than equal to 0 right. So here of course this is you know this is greater than equal to half x1 square plus let me be careful here let me say this is not T but this is T plus 1 so this is greater than equal to half x1 square plus x2 square because T plus 1 is greater than equal to 1. So therefore this is greater than equal to half x1 square plus x2 square so implies V is again radially unbounded okay. So we can also have let's see let me try to think we can also have some kind of oscillating examples like the one that we showed in the picture and that can be constructed say by again introducing a time variable as 1 plus say sine square T divided by 2 x1 squared plus x2 squared right. So this is again greater than equal to half x1 squared plus x2 squared because sine square T this is of course greater than equal to 0. So this is this is fine and this is radially unbounded but what this term is going to do right this what this term is expected to do is to sort of make things oscillate around a bit yeah make things oscillate around a bit. So this term is going to contribute to some oscillation but in time right here if you notice the oscillation was with the state itself right that can also be constructed right I am not doing it here right but this can that can also be constructed okay. So I would really urge you to give it a thought as to you know try to try to construct such a Vx okay try to construct such a Vx right where it's you know x and it's oscillating with respect to x itself but it's still dominating a class k function right still dominating a class k function it's it's not too difficult honestly speaking yeah it's not too difficult I mean you can think of I mean the simple ideas are like you know I can add an oscillatory signal to x squared yeah something simple like that yeah as long as I do that I'm also still okay all right okay so these are all examples of radial unbounded functions again relatively simple examples okay but again when things become more complicated we do require easier conditions just like before just like the case of positive definiteness we do require easier conditions to verify radial unboundedness also okay so what is it so again if I if I think of the case when V is not a function of state and remember that when sorry V is not an explicit function of time and remember whenever V is only a function of state we have been using Wx as the notation right do not get confused at all and using V and W almost interchangeably it's just the purpose of the function is what is important and the notation of the function itself so whenever V does not depend on time explicitly I am denoting it with Wx okay so whenever it's just a function of the state right then I have three requirements right first is that W0 has to be 0 Wx has to be strictly positive for all non-zero states again I'll come to the difference between the positive definiteness case soon and finally Wx has to go to infinity as the state goes to infinity okay so this is important it has to go to infinity in any possible direction it has to go to infinity any possible direction so what is the difference first of all let's try to see what's the difference from the easier condition for positive definiteness okay so let's keep this in mind yeah and and remember that this third condition did not even exist for positive definiteness there was no requirement for the function to go to infinity at all so obviously this third condition is completely distinct right so we are not even going to discuss it let's look at the first two conditions only yeah if you look at the first two conditions you require the same first condition that the function itself be 0 when the state is 0 there's no difference there this is pretty much uniformed along all definitions concerning Lyapunov functions yeah that the function needs to be 0 at 0 why because 0 is where we are interested to go to right so think of the Lyapunov functions or these v functions as some kind of energy functions right and we want the system to settle at the origin that is x equal to 0 therefore we want the energy the so-called energy or this notional energy function to also be 0 or minimum value at 0 at the origin right doesn't make sense otherwise right yeah if it's not at the minimum value when it's at the origin then it could possibly go lower right and therefore we will move from the origin doesn't make sense since we want to stay at the origin right so all these functions invariably require that they be 0 when the state is sealed so this condition is identical no difference second condition is where there is a small change it is positive for all x but in a ball removing the origin it only has to hold true in a ball okay of course removing the origin but here we need it to happen for all rn this is also because how radial and boundedness is defined right that that this domination has to happen for all rn right everything has to happen for all rn because we are talking global notions here okay so this is the difference so the first two conditions I find me I would like to call it some kind of global global positive definiteness it is still positive definiteness the only thing is we have we are requiring something more stringent that this positive difference hold for every value of state other than the origin and not just in a ball around the origin okay so this is a little bit more stringent and beyond that we require that the function goes to infinity as the state goes to infinity so what we want to do to look at this you know sort of easier condition thing is to look at a lot of counter examples right so and that really helps us to understand what is not a really unbounded function right so I am going to label this counter examples okay and 1 v sorry v x1 x2 is equal to half x1 plus x2 squared okay so the question is is this positive definite is this radially unbounded okay so we have to answer all these questions so the first thing that you need to notice is that along x2 equals minus x1 what is this if I look at the state space I am going to draw the state space right this is x1 this is x2 ah you know why that's happening right this is x1 this is x2 and what is x2 is minus x1 I believe it is something like a line which looks like this ah this is this x2 is minus x1 is this line right so this was x2 equal to minus x1 so what happens along this line along this line so I cannot of course draw it because it's I like to do it in three dimensions v takes zero value okay so v equal to zero all along this line okay and that is a problem right that is a problem right why because since v not greater than zero for all x1 x2 not equal to zero in fact there is a line which can go all the way from zero to infinity and minus infinity where v is in fact not positive at all it's in fact exactly zero therefore v is not even satisfying the first two conditions right so this is I you can you can very safely say that not positive definitely when yes so forget radial unboundedness this function is not even positive definite okay so definitely doesn't work next counter example so I'm sorry I should have used w but okay that's fine I've used v but again let's remember we are using v and w interchangeably and it's the purpose that it serves is what is valuable to us and not the notation okay so let's not get too confused or too hung up with the notation okay all right so let's look at v but w now I will use w right x1 x2 is equal to half x1 square plus one fourth x14 okay is this positive definite is it satisfying so of course w zero is zero great okay but what happens let's see w of all zero comma alpha is also zero because notice that is no x2 here only one state appears so whenever you have a function which has only one state what happens I can just say w zero alpha is equal to zero which implies w not positive definite and if it's not positive definite implies w not radially unbounded yeah because I I need minimum positive definiteness to even go to radial unboundedness okay so no answer is no again okay so if you have any function which has only some of the states appearing and not all of the states it is immediately not a positive definite function and therefore not radially unbounded okay so you don't even have to do any calculation okay because you can always make an argument like I made here okay great let's look at a slightly better example w x1 x2 well I let me use something better we have already seen this example w x yeah where x can be in rn yeah so what about this does this satisfy the first two conditions one is is zero it's zero yes w zero is in fact zero yeah what else w x is in fact positive for all x not equal to zero it's obvious because if x is not zero norm x is positive we already made this argument so so this immediately implies norm x is positive and I'm done right if norm x is positive right then this is positive this is again some positive denominator so it doesn't matter so w x is positive so of course it satisfies the first two conditions great yeah excellent we are happy at least we have something that's positive definite right which we already proved had did prove in the previous lecture okay so this is not a different example what about radial unbounded that is what about the third condition does it go to infinity as x goes to infinity as you can see the answer is no right why this is where I had written a sort of expression for w last time this is where it will come to be of use to us right so w x can be written as 1 minus 1 over 1 plus norm x square okay so as x goes to infinity w x goes to 1 right because as x goes to infinity this goes to infinity so this whole thing goes to 0 and so on I'm left with this one so this is one of those functions which is a nice positive definite function so so from here I of course had this is this positive definite but because of this implies w not radially unbounded iub is the notation for radial unboundedness okay so it's a positive definite function but it is not radially unbounded because it doesn't go to infinity as the state goes to infinity okay so this is rather critical okay rather critical distinction so function like this cannot be used to prove global stability okay all right it will not help us do global stability properties okay then we'll see later on of course but yeah so this is not a radially unbounded function all right excellent so all right these are sort of the you know the easier condition corresponding to radial unboundedness the second condition is when we have a function of both state and time I think both time and state appear exactly you know exact parallel to what we did for positive definiteness we had one condition for when we had a function of the state only and another condition when we had a function of both the state and the time explicitly all right in such cases we just use the previous easier condition right we just again just like positive definiteness we just use the previous condition what do we say it still has to be zero for zero state values and all time okay um and but we said d in r plus here of course so fine that's non-negative time I mean anyway to be honest just to prevent ambiguity which better to say it is r plus yeah for all definitions of v time in r plus is quite okay okay because we never deal with non-negative with negative time okay so we have v t zero equal to zero is of obviously our standard requirement but the next requirement is simply you know using our previous positive radial unboundedness definition is that this v tx has to dominate a radially unbounded function of the state only okay v tx has to dominate wx which is radial unbound which is a radially unbounded function of the state only where wx is radially unbounded okay so wx is a radially unbounded function of the state only okay in this case of course constructing examples are really easy I can simply take a v v tx as again something like t plus 1 by 2 and norm x squared and we know that this is greater than equal to half norm x squared which is of course radially unbounded so this implies that this is radially unbounded okay so it's not very difficult to construct such radial unbounded examples of both state and time all right great great so what did we look at today we essentially did the exact same thing that we did for positive definiteness or the case of radial unboundedness which are functions which addition in addition to positive definiteness globally also go to infinity as the state goes to infinity we looked at some easier conditions to verify this radial unboundedness also right so again of course we will continue with the next set of functions in the upcoming lectures so that's it thank you for joining


What's going on everybody got Kyle and Sean here late sports cards We're ripping 2023 bow and chrome baseball choice 12 box case number 27 random team style didn't randomize for the breakout Matthew B With the Reds down to down to Ken H with the brakes already guys. Good luck. This is 12 box full case 27 We'll get a Cosmic full case going next Monster come on Shawn wants a monster. Let's do it come on Shawn wants a monster Fox one. Okay, everybody. You love everybody. I'm down. I'm down Okay, uh cosmic full case going next Zach wasn't ready Pete Makes it a water a little bit. Yeah, maybe makes it a water. There's a uh Johnny Perone first. All right, uh, um Prospect Otto Johnny Perone. I was gonna say first. He's not first race Johnny Perone prospect Otto Johnny got a base auto for the white socks Dario Borrero white socks first And refractor Otto is going to be a high-year Casanova a gyre to 499 First refractor for the brakes That's number two Zach Wilson MVP overreaction Monday. That's uh, it's a thing God, do we got to do man? Yeah Honestly, if he didn't drop that uh, if he didn't drop that snap man, they had a good chance They had a very good chance. I was uh, I was a little worried there honestly. He played pretty well He did Wilson to Wilson connections a pretty good one too, man Nice, we got a um base auto of felon then celestine. There we go for the mariners Mac that for your mariners very nice hit there Felon then celestine. That's Jonathan with the mariners Top prospects in this one Got a reds first woman base auto of trey faulting trey faulting And purple is going to be corner stained a 250 first woman for the Rockies Little color match Rockies there on the purple box three. Good luck everybody Patty Mahomes is trash like Sean. He's pretty good He's pretty good Another nice one here for the Yankees gonna be Brandon Maiaa a nice one on the first woman base auto That's Jeffrey with the Yankees Nice Maiaa there We've got a um, this one's going to be a choice to 150 for the Astros Jacob Melton On the choice number to 150 So you can sign all those cards at like an angle like how does he uh, frenzyk been uh, been cosme to 299 speckle first moment. That one's going to be the uh, oh Orioles true I'm allowed to go. It's time to be straight Fox four box four Yeah, it was already yeah, we did see it saying it was uh someone posted on saturday night. I think Saw a picture like right when I woke up in the morning on sunday Here's a another nice one. It's going to be a first woman base auto of Rainer Arias for the Giants Doing well on some top prospects here Got a gold this one's going to be for the D-Bax Christopher Torrin on the gold wave first for the D-Bax You go, um Zona And Spence Kaufman got to be speckled to 299 first for the Padres Spence Kaufman Yeah, that's a clean. That's clean. That's clean Yeah, so you can keep doing it You gotta sign uh 499 another refractor that I don't know man Solace out of 10 Downsaint down saint Get spicy for uh subscribing I'm down You know me saint. You know me. Oh say always down See which way they're going this one is going to be Rockies brine bettencourt first woman base auto bettencourt Got another trey faultine first woman base auto for the Reds And to 499 refractor first woman auto Brett wisely rates You had solace in the 10 case really Wow, did we uh did we hit Hit nine mags hit nine mags. Let's go saints. That's awesome. That is awesome for sure That's sick Some really good spots in there even the um even some of the non the non name spots are pretty good man. You gotta You gotta bring truck full of cards First for the Cubs is a Don Sanchez first woman base auto Got a Simro Berset base auto for the Blue Jays And yellow is going to be Sarmiento for the Metz to 75 D'Angelo Sarmiento on the yellow to 75 All right, you guys halfway box seven Hit some good names so far. Let's see some more color though Some really good names so far my a arias and a celestine Too much uh too much talking There's a first hazers galiz that one's Dodgers first woman base auto Kind of we have green to 99 this one's green lava of Hayden Dunhurst For the Royals green lava to 99 And another yellow to 75 this one is Jesus galiz Dodgers We'll be as disconnected reconnected reconnect Reconnect all right box number eight box number eight love it. Love it Yeah guys 11 to go on our next break is going to be a select basketball six boxer random team style Is our next break with spots appreciate every phone us up so far keep on moving here now Got some breaks filled up and keep on going Here's a Padres Eddie Beltray on the first woman base Eddie Beltray There is um christian manna for the white socks another first woman base auto And this one is going to be Volpe on the bowman ascensions insert rookie Otto nice one there Yankees Mr. Jeffrey that's a sick looking card there to 150 Um, we'll send a box with this Jeffrey. You can almost see his uh auto on the back there must have set it on a uh Card that was too wet there for sure So we'll send a box it is that's kind of crazy. You can almost see the Volpe So we'll send a box with this one Jeffrey for the um smudge on the auto I believe there's something can get that off, but we ain't going to be doing nothing with it. So We'll send a um box with there All righty guys box number nine Keep it going appreciate everybody buying the spot so far today keeping me and Sean busy here Thank you Eric buying into our mixer Thank you Jeffrey buying a spot in select and um Jungmen buying the cardinals in one football. Thank you Yeah, just hit refresh Nicholas if it crashed for a second there it uh, it came right back Here's another one Rainer Arias for the Giants second Rainer Arias base auto nice one Giants Second Rainer Arias. We've got a Simra bear say first for the uh blue jace first for the blue jace And to 499 refractor first up treyfal team reds Yeah, pretty much a dual auto slap pretty much box 10 And guys we'll get a full case of cosmic going next One is division style A you'll see galon first woman base auto for the Rangers Got a Demetrio croissant days d backs first and on the back. We have Javier revas green to 99 for the pirates first bowman base Hey, let's go spicy. That's awesome Red Lewis Hamilton to five we will take that Absolutely, man. That's fire Yeah, we had a good deal on the f1 sapphire yesterday. I think it's still the same price had it down to 199 yesterday Dig in the f1 man. I'm a big uh like the f1 cards Need to get some dynasty, but those are expensive. I need to get one or two though There's a first for the royals junior marine Royals first bowman auto it's like yellow on the back. This one is riccardo pares first for the Phillies Carter Perez and yellow is going to be for the pirates the 75 junsock shim first yellow auto pirates And last box here number 12 Appreciate everybody nice on the front. We got Ethan solace first bowman base auto for the Padres There we go Padres. We'll mag that for you Padres. That is going to be Andrew Andrew M Ethan solace on the first two arias is a celestine a solace Nice case of choice Griff McGarry on the first for the Phillies and on the back. We had a luke berry hill refractor first Um for the astros do 499 All right. All right guys that will do it for bowman chrome choice case number 27 A quick recap here. We had pares marine croissantes galan roberce One arias had maina Eddie Beltry first Padres. This one was galees Sanchez Um roberce again. This one was falting for the reds Betancourt is another arias for the giants had a brain in my a yankees falting Borreiro prerone There is the 499 a Casanova Um the 499 we had wisely and falting had griff mcgary Um, this one was 499 luke berry hill to 299 Kauffman and benz Cosme This is the 250 conor stain. We had um to 150 vulpe on the bowman ascensions We'll have to send a box at that jeffrey. There was some uh some ink got over on that one got melton to 150 To 99 revos and dunhurst to 75 galees and shim and star miento to 75 gold is torrent Then we had two mags. We had a ethans solace base first bowman auto and we had a felman seliston for the mariners base first bowman auto Nice one there guys very good, uh case of choice some really good prospects in there That'll do it and we will get some cosmic going next. Thank you guys


All right, cyber traders welcome on back here to close things out for the week this Friday, February 26 shortly after 2 30 p.m Eastern time welcome on back here for our afternoon meeting as I just said Let's get a quick sound check just to begin make sure everybody can hear me loud and clear I want to hop right into the thick of things we ended up seeing this FSR trade and break a new all-time high Of course here once again today great to see Jean Ron Michael Stefan Laurie and a quick shout out actually to all of Our new gold students joining us here today. We have Craig D We have Dan M and Daniel B all joining us here as new students So welcome aboard folks and know if we're always streaming on social media welcome on back So, you know this FSR trade ended up making a fantastic pop yesterday And I tried recording it pretty much at the same time as I was doing this meeting and then afterwards and we didn't really see it Make as fantastic of a pop as we see here now So allow me just to put the Dow chart up for the meantime Dow dropped off after the market had opened up but FSR Man this thing after coiling up, I guess you could say underneath 2250 Thing just took off right away after the last breakout going into like 1115 1130 Eastern time So man just broken all-time high made the move that we would have hoped for yesterday and with that This is pushing right back up So for the afternoon certainly we're just gonna put this on our main watch list I just want to go over this channel pretty quickly ended up building this really nice channel off these lows and highs and You know for what it's worth the volume died off. So if it's not gonna break out any further It's just gonna break down below So this is actually a pretty good short opportunity as soon as that broke that channel I was in for a very small position a hundred shares just playing the waters on a very Low volume Friday. Otherwise, I'd say FSR not low volume But nonetheless just ended up making nice bounce roughly around where I got out 2590 was my exit Ended up essentially holding that 26 moved right back on up now So yeah, the FSR trade definitely gonna put back on our watch list more so for the reason that it's moving back up So I don't expect this to like necessarily drop back down to like 25 or 23 like I was hoping for once it broke the channel I ended up realizing it was building support. So I had to fold my cards pretty quickly there So yeah for right now, it's moving right back on up see if this can break into all-time high now I will say I'm not recording this trade right now. I wish I was from earlier But for all of us in the phase 2 stock course for all this inside the trading room that have access to the phase 2 course We're gonna be talking about conveniently enough all-time highs and lows So, you know, we tried recording the stock yesterday as it was first breaking out and it didn't really make as clean of a move We'll talk about it and then from here. We'll discuss at least the big pop from the mid-morning and into right now So let's just go through folks quickly from the top percent gainers list through the losers list see what else we got the FSR We can let's push right up to the high 2780 So the FSR trade was beautiful right beautiful trend provided good opportunity as a short on on the break of that channel This is the complete opposite of what FSR is this is like obviously moving up But it is just as choppy as could be it's very difficult to read minute by minute Liquidity is alright penny spread, you know decent order book in terms of the size of a lot of the orders it's easier to acknowledge support resistance at some of these prices, but That's what you would like to think by reading the matrix there Otherwise, it's been a very difficult stock to try and anticipate Anticipate clean breakout trades on so don't get me wrong and squeezing back up You know once again and the high the days off 658 so it pretty much has like another 90 cents to go right now I don't mind but putting this on the side for the meantime. I probably wouldn't be surprised or I won't be surprised I should say if I put this on our main watch list just at the end But I want to see if there's anything else just looking better setting up for an easier opportunity perhaps Sorry about that FSR warrants number two We're not going to focus on the warrants as we are going to focus on the equity LIXT here. Wow, this ended up making a big push in pre-market But as soon as the market opened up right the same with this as we see with a lot of these you know small cap Pops and pre-market ended up dumping off right away never really set itself up for a good move back up after Ended up diving down. I guess here to the low at the time. We caught this during class I thought we were looking at this roughly around. It was more like here. Yeah, like 535 537 So it ended up actually breaking above that price after made for a clean breakout from where we left off with it in The phase two course But ended up actually dying off from that point like peaked up towards just about 615 620 a shear and then then then pull back So yeah, right now just not gonna be too focused on this INTZ the FSR obviously on our list already It's ready at 28 breaking a new all-time high right now I don't want to fumble around and try and set up another platform trying to record I'm gonna miss the move and just take too long here So we'll talk about that trade folks coming up in the class the AFI trade This is one I was talking about from the class this morning in fact and very briefly It's not gonna be like really that tradable right now But I was really looking for a a gap fill on the way down here this morning It's shortable whereas the LIXT was not so whereas that dropped off and would have made for a great short This did drop off a little bit But ended up slowing down pretty fast pushed back up towards five actually made a pretty quick pot from there So I did all right on this trade as a short I think I picked up like 16 or 17 cents and then I ended up taking a couple small losses just before but You know right now. It's just pretty flat So we're not gonna focus on this trade for the afternoon Maybe keep it in mind for next week ended up making like a big gap down back You know a week or two a week or two ago and then you know funky gap up here So there's a lot of volatility with this trade. You just got to catch it on the right morning AUPI not much going on there OCG and there should be a lot going on here speaking about volatility and gaps and ranges You know the OCG and obviously we saw this going back into earlier this month made a huge push from like three all the way to six Gapped up all the way, you know towards just about a gap up to basically Nine and then pop right from there up towards a hive almost 20 So for what goes up comes back down the stock was shortable back then when it dropped off It was shortable this morning although there was no short on this trade ended up beginning to push right back up It broke above 850 resistance took off pretty quickly one of our students Norman Called this out to me in our coaching session earlier. We missed this fly all the way up towards just about 1055 I ended up doing better on the way down the stock was a great short Like I'd said from a few weeks ago as it pulled back down after the run up Well, same thing happens here ended up beginning to make lower highs pulled back down below 10 Ended up doing pretty good on this short from 10 here ended up squeezing all the way down just towards about 9 20 ish 9 25 and pushed back up so in terms of the tradability the volatility I should say we're gonna put this on our list definitely OC GM If it spreads ended up topping off just before here at 10 bucks. So easy to see level I know if I was so I was looking at this Je trade from earlier today filling the gap on the way up And we didn't talk about this one as much in class So, you know really quickly you could see on the daily chart notice and it did gap down after this day But there is a gap based off of this low So, you know 523 you could say, you know is the gap to fill based off this low and Here today pushed all the way up towards a high that fell fell 10 shut 10 cent short still 513 So, you know, you could still just say that it's just the fact that you wouldn't have expected the gap to be Based off of this like close or this low even no, I mean you had a day before where it's actually actually lower So that's how you would interpret that So a plug make a nice move today PLUG pushed up pretty nicely right before class at least popped up from like 45 just about up to 50 Check in on fuel cell FCE L while we're at it topped off I thought about just around 1750 earlier today I was looking for a re-entry on that trade or a re-break over 1750 if plug kept pushing So it was surprised now that both, you know, pretty flat way fair up 11% I guess be earnings on that, but I'm not sure I Guess I maybe for as much as they dropped off the day before maybe they had earnings earlier in the week I'm uncertain either way just pretty expensive just want to check in on the price Speaking of a pump-and-dump type move here rocket RKT ended up making a big push in pre-market I know that their CEO is on CNBC ended up making big squeeze up But came right back down and it's not to say that you would have expected the pullback after 930 I mean perhaps you would have seen a higher low for this to Re-break above a resistance off of like 23 bucks But it never even you know made that chin never made that move up until like 12 30 in the afternoon And you know at that point you would expect for it to be a resistance You wouldn't expect a clean like breakout after three and a half hours or two and a half hours of just you know Dropping off and sell it so yeah at this point right now It's just pretty weak, but I was looking for a move off of 22 a bit earlier I ended up taking like an 8 cent loss off of 22 from this false break here I was more hopeful to see the squeeze back up ended up stopping me out and at this point now Maybe it makes that move, but just it's just again end of the week. I'm not really too interested in that I want to see something trading a lot more volatility than this so far Big volume spike here. I don't know what the hell happened on this VTG and I just saw it on the five-minute chart right away Yeah, so on the tape you could just see from the last few minutes Just now pretty much 237 a slew of prints like cell prints here in red 89 K or 90 K just about 60 K just about a hundred forty three or 4k here So don't know what's going on at 237 looks like a lot of selling or perhaps short selling or perhaps even buying I'm just based on the reaction after here. I count that as cell prints So, you know if this ends up re-breaking under 237 or so then I think there's a chance to see this like really tank But you know for right now it's pushing back up pretty nicely right away So We'll see we'll kind of put this on our side list given that spike Otherwise, it's like as dead as can be and just pretty slow. It was pushing up a bit But this is the thing that interests me more right now All right, so let's hop over to the losers list right now folks markets down 338 points and counting so I'm sure that we're gonna find a little bit more on the losers list As that pushes up yet another penny. I think The REPH is shortable for trade station users here It's at the low of the day. I Like that so check to see if it's shortable from you if you don't have trade station if it's not then focus on something else from our list so far, but Yeah, right now. We don't really have too much to work with we have the FSR the OC GM This is shortable for right now ended up making a huge gap down on the daily chart Obviously pre-market activity included there But ended up like really making a little squeak up towards like 360 or oh, I guess that's four at first Oh, no, this was the market open. I'm looking at this peak here. I'm like is that for no 370 yet So I thought 370 this squeaked up to and then after that just felt fell flat So all I had this tour list on the way down I mean, you don't really find these stocks below five bucks to be shortable that often so this is a nice surprise Tdca not under five but pretty close and also shortable dropped off in the post market yesterday I don't know what the news is. We're just gonna conserve time for class, but Opposed as opposed to the REPH. This is like really flat Legitimately flat as can be right now the REPH I was saying before just kind of fell flat from the 370 resistance and then obviously now spiking down pretty nicely Well, what goes up comes back down the CDXC Fausto's move from yesterday that like I mean exploded Right and from ten bucks after the market opened up. We call this out right away. I mean we isn't him I don't want to take credit for that. That was all Fausto there So pushed all the way up towards just about a high of like 24 at the time I think and then dumped off but yeah, what goes up comes back down. This is shortable. It was shortable yesterday We were wondering when the drop-off is gonna be and we thought it was gonna be at some point today if not the morning, but I Just you know, hey, this is a level five stock I'd imagine just based on the spread it already dropped off so much already So the question you should start to be thinking about is at what point does it begin to just exhaust or even spike back up? It looks like a broke below is support off 17 or roughly 1750 perhaps from earlier now at 15 So, you know, hey, if you're a level five trader in our trading room, you know who you are You know, maybe you focus on this, but if we're just talking one over the other The REPH is cheaper has much tighter spreads. It's just easier to follow It looks a little bit flatter compared to the other one But looking at this drop off here across the last 20 minutes. It's not a bad move. It's a level one stock Definitely good stock to try and practice shorting on Up dropped another two cents. There we go. All right. What else we got here folks I think that just about wraps it up at least from my eye I don't really see much else that looks like a familiar name that would be tradable shortable up Right as I say that workhorse So workhorse I should have saw this right away rather but ended up dropping off I mean shortly after the open little pop and then drop back down So yeah, this one definitely gonna make on our watch list again again We're kind of not at a shortage, but we're not too overwhelmed with stocks on our list so far So the workhorse will we'll read on the way down? All right, John calling out the Viacom trade Viac at 6450 Nice recovery from yesterday's drop-off Liquid order book for a 64 dollar stock, you know, it's a good side list stock If you did well on it from earlier today or if you want to follow it then You know it looks decent like a setup right, you know holding that resistance pretty nicely Scott asking about the car not asking you saying cost moving up nicely rather Yes, biking up pretty quickly. This is a pretty good push-up. So how about this? I'm gonna combine two call outs right now because cost you could see going back until each January Popped up and down at the same time as GameStop did we have one of our members on YouTube right now asking about AMC So let's just check them out here GameStop AMC cost popped and dropped at the same time so Cost is getting a lot more of a pop than the others so far. It's more volatile. I should say at least but yeah AMC squeaking up a little bit. I mean, hey, I'm not going to be a hater just yet of AMC Although what my but yesterday I had a much better day today than yesterday at least but yesterday AMC What my but just not like big losses, but I picked the wrong direction So I took a bunch of these tiny losses over and over and that was actually right after I said I think from the meeting before back on Wednesday, I was like, oh, you know, I normally do good on AMC That's one of the stocks I do all right on so why not, you know, so listen, I'm not a total hater But right now I just think after this squeeze up a little more reluctant here just before it ends up potentially moving back Up again. So now let's see if this could break under an over eight And then if it could break under an over to eight hold from there and run up again So that would be the hope keep a good eye on GameStop otherwise just use that as like your like indicator You're like ETF almost so to say You know in the sense of just saying this is a much more volatile stock They're moving pretty similarly right now So if you see I go big pop on GameStop just wouldn't shock you to see a big pop on AMC John says one letter out from AMC Josh AMD was better. There we go. So let's jump to AMD at least from today Downs from 83 held off just about 85 50 essentially there And check out our lovely riot our Mara and then along with that I did want to drag up bitcoin going into the weekend for anyone streaming on social media I didn't have it up yesterday, but I want to just jump to it pretty quick because well if you recall back on I think it was Wednesday's trading just because yesterday we saw it push back up But Wednesday, I was concerned about this I was concerned about the fact that we saw a lot of chop around the 500 EMA on the one hour chart not the chop I was worried about but I was worried about these closes below the the uh the 500 EMA Well, right after we finished up that meeting like literally here on this bar at three o'clock eastern that day It's spiked right back up popped right back up Well all is well, right And not just yet. It's pulled back down across the Last evening heading into this morning Going into the open it bounced up back up a bit but heading into the weekend though Obviously thinking long term you're seeing a drop a lower high here and then a push back down So good news will of course spark a new move back up or some sort of pop But otherwise if there's no, you know event or no news coming out over the weekend No, like CEO or company dipping into bitcoin or making it official Then we might happen to see this begin to shake down even a bit more And that's where I'll kind of drag back my six hour chart that I had up originally from that day Wednesday this week So I was saying at the time that it was certainly possible to see this pulled down Should this drop off more from this EMA now on the uh six hour chart This is the 100 EMA on the six hour roughly around the same price as the 500 on the hour But you know, I think that perhaps if this continues shaking down That we could see this probably dropping down towards about 41 41,000 maybe this you know EMA here the uh 200 On the six hour chart. I like using the hour and six hour charts. That's my personal preference So to each their own there, you know, everyone's gonna have their own different charts different EMAs But at least as of now from from where the last time I I spoke about crypto bitcoin and showed you the charts You know, I pushed up a bit, but then now since Again, I move back down so there might be a bit more to come here All right, so just be aware with a marrow marrow and right in that regard too All right, folks So if there's anything else that's moving left right up or down, let me know if there's anything else that looks good Might I add Kathleen saying the ex pr also moving alongside with the amc the uh game stop Yep That one's a cheapy. So that's a good one to watch too right alongside amc Cost is still spiking up here a bit folks So, you know, that might be the more active of all of these so far heading into the afternoon I'll add this to our main list If you've done well on this trade in the past Then I would say for you folks to look to trade it for anyone else That's not joining me in class and has yet to trade the stock or is new There's plenty of other stocks to trade out there and those that are on our list here Like again, I can't stress enough the re ph Well, hey, it actually spiked right back up after we uh last left off with it here So I may have jinxed it but you know, this stock still has great tradeability in the sense that it has a penny spread It's down big it's shortable. So, you know, if it pulls back down here We might we might end up seeing a lower low Otherwise, we might see a bit of a recovery a dead cat bounce as we talked about from the morning class All right, so, uh, you know folks if there's anything else here, just uh, let me know Oh, if you missed the bitcoin chart for all this live inside the trading room I'm going to copy and paste I probably uh screen projected the wrong way or not the wrong way But just differently so I'll copy and paste for all of us inside the live trading room there. Don't worry Do that right away here. All right, folks so for all of us in social media first and foremost enjoy your weekend as always it's going to be nice and uh Sunny out here in Long Island for for for the first time in a while. So be a nice change of pace here Step outside right after the market closes. Enjoy. Enjoy. All right So folks, um, aside from that though if you're interested in learning more about cyber trading university Just make sure that you like subscribe notify right on our youtube channel youtube.com slash cyber trading you Otherwise feel free to join us. Uh, go to the link right down on the banner below to sign up to join us inside our trading room For for all of us inside cyber group. We're going to post the link for class We're going to post the picks of course coming up here right away after and aside from that I look forward to seeing most of us inside the phase two course later on here. Take care


This is stock number 9986. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used heavy-duty truck headquarters. Today we are checking out this super clean 2014 Ford F-250 Crew Cab Short Box Lariat. This truck has the 6.7 liter power stroke diesel. From this HD video you will be able to tell that this truck is extremely clean all the way around inside and out. Headlight lenses are nice and clear. Some nice fender flares on there. We shoot all of our videos in 1080p so if you have HD video, HD capabilities on your computer tablet or smartphone, turn them on right now because it's like you're right here looking at the truck with me and definitely your best way to check out the vehicle without before you actually get here. Comes with XD Series painted alloy rims. Open Country Toyo MT, 37 by 13.5 R20 tires. A little under half the tread left. Framing under body looks like it's in really nice clean condition. Sterling gray metallic is the color. Front bumper is in excellent shape. Didn't see any denser dings on that. And the passenger side fender is in really nice condition as well. Passenger side rim, no scuffs or scrapes on that. Take a look at the big lift on here. And as you go down this side of the truck you can see just how clean the body is, how reflective and mirror like the paint is. And that's why we take the HD videos so you can still see the truck here. The truck and of confidence in the vehicle, whether you're close by or far away, has the chrome tipped exhaust. Cab is in excellent shape and very, very clean down this side. Frame and under body, very little to no corrosion. Looks like it has been straight piped and leaded. Back rims in excellent condition and the back tires have just as much tread as the front tires. FX4 off-road package gives you the skid plates as we come around to the back of the truck. Rear bumper parking sensors, full towing package which includes receiver hitch four pin and seven pin wiring. Tailgate is in excellent shape, has a spray and bed liner, a line X spray and bed liner, gooseneck hitch, seven pin wiring. And it has the tailgate step assist right there as well. Tailgate's nice and easy to lift up. And on this side of the truck just as clean as that passenger side. And for full disclosure, this back rim is in the excellent shape as well. Very clean, does have the driver's side door code entry. We do have the code for that, the power scope mirrors.  There are no rips, there are no tears on the seats. Very nice condition, does have memory driver seat, power windows, power locks and those power folding mirrors they go in like so. And then they also power telescope out like that. Has the sony sound system, auto headlamps, power pedals, weather tech floor mats throughout. Inside you can see that this one has 49,753 miles. Instrument cluster is very nice and clean. It has the leather wrapped steering wheel, audio controls and bluetooth controls on the right, cruise controls on the left. It does have the six-speed automatic transmission, turned-off four-wheel drive, stability control, downhill assist control. Has the my Ford Touch radio, has the factory navigation system. This is also where your backup camera shows up.  You can also do your dual climate control there. Factory brake controller, four AUG switches and a 110 volt, 150 watt plug-in. Passenger seat is very clean as well, no rips or tears on that. Does have side curtain airbags. The headliner is very nice and clean. You have your power slide mirror window as well as your map lights right there. And we'll take a quick look at the back seats and then check out under the hood. It does have a live wire TS programmer on it as well that comes with the truck. Back seats are just as clean as the front seats. No rips or tears back here. It does have the power slide mirror window with the built-in rear defrost and we do have those weather tech floor mats back here as well. Has a 110 volt, 150 watt plug-in and the lockable storage compartment underneath the rear seats. We'll take a quick look under the hood. I would personally like to thank you for checking out the video today. Hopefully from this HD video you will have been able to tell just how clean this truck is all the way around inside and out. Under the hood we have the 6.7 liter power stroke diesel engine base very clean runs very smooth. This truck has been fully safe and inspected by our service shop. It has a fresh oil and filter change. All the fluids have been checked and topped off. The truck has been gone through mechanically 100% and is 100% ready to go. To see more pictures of this truck or one of our other 450 new and used cars, trucks, SUVs, minivans, Wranglers, half tons, three quarter tons, one tons. You name what we got it go to our website www.summitauto.com. Full pictures and descriptions of every single used vehicle that we have all at summitauto.com from two locations Beaver Dam and Fond du Lac, Wisconsin. And if you'd like to check out more HD videos you can go to youtube.com slash summit auto. Remember to like, subscribe, and share on this video and all the videos that you see there. In fact, in a second you will see a link to subscribe to our YouTube channel on your left. A link to more heavy duty truck videos like this one on your right. If you have not been to our website on the bottom link to this vehicle on our website click those check us out. We really look forward to helping you with this Super Clean 2014 Ford F250 Crew Cab Short Box Lariat FX4. Thanks again.


What's up everybody, Jason here for jaspyscagebrakes.com guys. Let's chase some Brock Purdy, right? Give it away a thousand dollars if we hit a Brock Purdy in this break guys. So this is 2022 Mosaic Choice, two blocks break, pick your teams number one. So again guys, two huge giveaways. First one is we're giving away the 49ers. All you have to do is buy a team in this break to get entered and have a chance to win the 49ers for free. There was teams as little as $9.99 guys. So the more teams you bought, the better odds you had to win the 49ers. Now the second giveaway is if we hit a Brock Purdy in this two box break. I mean the Brock Purdy is going to go to the 49ers winner, so that's already a dub. But as I added bonus, we're going to give away a thousand dollars back to one customer who bought into this break. So imagine if it's a customer that only paid $9.99. Could happen guys. Let's hopefully hit both of those in this two box break here guys. So dice roll here will be for the list of customer names. So I'm going to gather up all the names here, put them in a randomizer, delete the extra space there. And one person is going to win the 49ers and it's a six and a three, nine times, nine times gilag. One, two, three, four, five, six, seven, eight and nine. Nine times, six, three, nine, Dan, congratulations. After nine times you just won the 49ers. So that's the first step right there is winning the 49ers. Now remember it has to be an autograph. We get a party numbered card, birdie color. I mean great for the 49ers, they get that regardless but if you want to hit the thousand dollar bounty we have to hit this Brock Purdy autograph basically or any autograph Brock Purdy honestly. So let's see. Let's print out this sheet and here we go guys. Got two boxes from the back. We actually have pretty much a whole case worth in the back so I'm assuming someone opened up a case to sell as a personal in the store or something like that so we'll see what we can get here guys. And it's funny because we've done these before in the past and I remember like probably like in break two or three. We had like a Kobe Bryant give away, autograph give away and we freaking hit one. Hopefully rare but regardless guys obviously this is 2022 Mosaic Choice. I mean you're still looking at potentially you know some big candy pickets in here. You know Steelers have pickens as well, big time receivers like Olavi like Garrett Wilson with the Jets. You know amongst other big names with autographs and especially veterans so hopefully we still get some nice hits though. We do have a little kid reporter there. Alright here we go. That'd be wild if we hit this right now. Aw man we got some points. Come on man. Points, this is going to be a lot of points I think like 900, 1500. Wow 1500 points that's a lot actually. Someone's going to win that at the end guys. Drew Brees, number to 80, Taysom Hill that's another one for the Saints. We're going to Omar to 80 and then we got Arnold for the Atlanta Falcons and a Trion Davis Price for the 49ers. Alright well 15 points actually. Probably sells really well usually about like 150 to 250 sells for about 7 to 10 bucks so I'd assume that this is probably, probably close to like 100 bucks right? Maybe more. So we'll see who gets that. Oh there's a quick one guys so there's the next one. There's a Malik Willis. What? Are you kidding me? More points. What is going on here? Wow. What kind of BS was that? Out of 80 Jeffrey Simmons for the Titans, we got Luke Keekley to 80 for the Carolina Panthers. Aaron, then Jake Ferguson actually is a really good head of 80 for the Cowboys, Chris Parent. Then we got Carson Strong and then Jake Ferguson. Man well at least we probably got rid of the points boxes. There's no way we're going to get more of these anymore. Jesus that was horrible. Sorry guys well that's out of my control obviously. Dan Panini. But like I said we do have plenty of boxes pretty much a whole case worth so I expect to hit a party man. I'm pulling for one. And obviously just bigger rookie autographs in general. Unfortunately these two boxes have points so I can't really do anything about that. But the 1500 points is a nice one though. Alright so let's put that right there. And we'll do number one gets the 1500 points. Number two gets the 900 points. And like I said number two is on the website guys. You guys want to try to run it back. Definitely feel like we'll definitely get better boxes than that. There you go. Alright so let's switch scenes. And let's do the giveaway for those points. So everybody gets a shot at them guys. Just going to copy the names. And we'll get a new dice roll. Click roll. Three times two and a one. One. Two. Three. Three times three times. Jason K you're going to get 900 points. And Omar you're going to get 1500 points. So there you go. Top two names there. Get a little extra. We'll get the hits. I appreciate it guys. Like I said the next one in the store is already at five left. Let's get that one sold out. Definitely expect it to be much better guys. Appreciate it at jaspyscasebricks.com.


Welcome to module 38 of point set biology part 1. So far we have studied connectivity, path connectivity, local connectivity, local path connectivity and so on. So it is time to take up another very important concept, compactness. As a tagging along, the lindelofness can also be studied, which is not all that important for us and quite often just like you study connectivity and path connectivity to the compactness and lindelofness, go hand in hand. However, from the point of view of point set biology compactness is so important, if you try to tell the story of compactness by which I mean how the concept developed and so on, it will be half of the story of point set biology. So, so important in the compactness concept. I will not be able to do historical any remarks and so on because quite often they can be done only in the perspective, only after knowing what we wanted to do, what has been done and so on. Then why this one, why this one and so on, then you can go back and dig up history, historical development motivations and so on. So, let us start studying compactness in its earnest. Start with the topological space X. A subset of X, so let us say A will be called compact if the following happens. Suppose you cover A by open subsets of X, some arbitrary cover, then it must admit a finite sub cover. So, let me define all these things carefully, sub cover, open cover, what is the meaning of finite sub cover and so on. So, a family UI of subsets of X is called a cover. This I think we have defined earlier, but let us recall. If A is contained inside the union of this family, it is called an open cover. If every member of this family, this all these UIs are open. So, one can have closer coverings also, they seem to be not so important. By a sub cover of a family, for a given set, first of all it must be a cover, then you take a sub family of UIs, which will be also a cover for the same set A, such a thing will be called sub cover.  So, once we have these definitions, a subset is called compact if every open cover of A has a finite sub cover for it. Likewise, we say A is lindle of, if every open cover of A has a countable sub cover for it. Deliberately I have not put compact space and compact set and so on. Whatever you want to call it, if you want to think of this as a subspace, you are welcome. If you are think of this as set, we are welcome, compact set or compact subspace. What I have defined is a compactness here. Similarly, lindle ofness here. Okay. However, you may wonder why the subsets? Suppose this A is the whole of X, then it makes sense. All these definitions make sense. Then we may say that X is a compact you know, space by itself. No subspace, no subset. A compact space. So, its own topology take an open cover, any open cover, you can extract a finite sub cover. And you may not be able to extract. The definition gives you a finite sub cover out of it. That is the whole point. So, that is what I have repeated here. Namely, you can call it a topological space compact or lindle of space if it is compact lindle of as a subset of itself. Okay. It is immediate that a subset A contained in that X is compact or lindle of if and only if it is compact or lindle of as a subspace. You take subspace topology, then as a space, it must be compact and lindle of these two are equivalent because you have an open cover, open subsets coming from the larger space, intersect it with the given set, they will cover and those are the open subsets into subspace. So, these two notions are same. There is no difference between them. In particular, you will see that once a space is compact, it does not matter where it is contained inside as a subspace. Suppose X is a compact space. Suppose X is contained inside Y as a subspace or Z as a subspace, both the case is it will be still compact subspace. Now, if you take a closed subspace of a compact space, automatically it will be compact. Similarly, if you take a closed space of a lindle of space, automatically it will be lindle of. So, how does it get it? Just add one extra open set namely the complement of the closed set. That will become a cover for the whole space. From that, you can get a countable or finite sub cover. Now, throw away the extra thing, you do not need it to cover it. So, you have still countable sub cover or a finite sub cover. So, that is all the trick here. By extending the given cover for the subspace which is a closed subspace by putting one extra element namely the complement of that set. And then you can come back. Let X be a topological space and B be a base for X. Then X is compact respectively lindle of. See quite often whatever I do for compactness, I can put it inside a bracket lindle of, but not always. When it is not, I will mention that. If and only if every cover for X admits a finite, but every cover of it from members of B admits a finite cover, sub cover. You see, in the definition of compactness, you want to have every open cover should admit a finite sub cover. But here is a restricted thing here. Members of B cover it. They do not give all open covers. But if it is satisfied, condition is satisfied for members of B which is a subclass of open covers. Suppose you take a covering of X with only members of B and they admit a finite sub cover. That is enough is what this theorem says, this lemma says. So, this is the role of base to cut down our tile. You do not have to check it for all open covers. You have to just check it for covers coming from members of B. This is the gist of this lemma here. The proof is very easy. One way is clear. Namely, if you take a cover out of B, that will be also an open cover in the general case. So, it must admit a finite sub cover. Now, suppose this happens for only members of open covers coming from members of B, then why it should be true for any general covering? That is what you have to start. So, let the condition hold and Uj be an open cover for X, not necessarily from members of B. Then for each X in X, we have X must be inside one of the open sets here, Uj X depends upon X. But then, B is a base means what? There must be an element BX inside B, such that X is inside BX contained inside Uj X. Now, if you will vary all the X, that will cover, this BX is will cover X. Now, by this condition, there will be a finite cover out of these BX. Let us call them as BX1, BX2, BXN. But each BXI is contained inside the corresponding Uj XI. Therefore, X is contained inside union BXI, contained inside union of JXI. So, these were the members from the original cover. If you write and replace n by infinity here, you will get lindelofness. So, proof of, for the lindelof property is also the same thing here. The next immediate thing is the following. Under a continuous map, compactness and lindelofness are preserved. What is the meaning of this? F from X to Y is a continuous map. X is compact, implies FX is compact. X is lindelof, implies FX is lindelof. Just like connectivity, local or path connectivity, etc., not local connectivity. Connectivity and path connectivity we are seeing. What does that mean? Immediately it means that under homeomorphisms, compactness and lindelofness are preserved. In other words, they are topological properties now, in our formal definition.  Suppose A is compact respectively, lindelof and we have a surjective function F from A tau A to B tau prime B. This tau restricted to A, tau prime restricted to B, they are subspace topologies. And this is a continuous function here. You do not need F to be defined on the whole of X and Y, whole of X to Y. We have to show that B is compact, if A is compact, respectively lindelof. So, how do you do that? Very easy. Start with an open cover for B. F inverse of UI intersection B. See, UIs are now open subsets in tau prime. And all that you have is B is contained in the union of UIs. But if you intersect UI intersection B, this is an open subset of the subspace topology. F is continuous. F inverse will be open inside A. And since this is a covering, F inverse of all this union is the whole of A, which is same thing as union of all F inverse of UI intersection B. So, I have got an open cover for A.  What do you have to do? What are the, what are the meaning of these are open subsets? Each open subset F inverse of UI intersection B is some UI intersection A where UI is open in X. In any case, these VIs will now cover A because they are obviously larger than the original space here. A itself is contained here. So, A will be contained here also. Therefore, there is a finite respectively countable subset I contained inside J. Subset A is contained inside the union of VI, I over I. This is by compactness or little delofeness. But now you can come back because these things are larger than F inverse of UI intersection B. So, B will be contained inside the corresponding UI's which is either finite cover or a countable. So, that is the consequence of this elementary result namely under continuous functions compactness is preserved. Therefore, it is a topological property. In particular, it follows that compactness of a subset A of topological spaces does not depend on how A is sitting inside X. I am repeating it again. Once A is homomorphic to another A prime, A and A prime are sitting wherever they like, okay, inside the subspaces. If one is compact, the other one is compact, over. This is the same thing with little delofeness, connectivity, path connectivity, etc., that we have studied. They are all topological property, that is the whole idea. Now there is a partial converse here. Converse means what? Compactness from the domain to core domain we have seen. Now we want to count the other way around. Suppose you have a surjective open mapping. In particular, it will be quotient also, but not just a quotient, surjective open mapping. If Y is compact and each fiber is compact, then X is compact. So, this is similar to connectivity wherein we had just a surjective quotient map only, not necessarily open map. Open surjective map is a quotient, but not the converse. So, weaker condition, we had similar result for connectivity under weaker condition. But now for compactness we have put a stronger condition, surjective open mapping. You will see why this is true in the proof, why we needed open set here, open mapping here, okay. Right now don't worry about why it doesn't work for quotient, just quotients and so on. So, let us work out this one. Take an open cover for X, we have to extract a finite sub cover. For each Y in Y, choose a sub cover U, Y1, U, Yk, Y for Q inverse of Y. The Q inverse of Y is given to be compact, a compact subset of X and U is an open cover for the whole of X. So, it will be a cover for Q inverse Y also, okay. So, now I can take a cover for finite cover for Q inverse Y, okay. For each fiber you get a cover, number of elements here will depend upon Y, okay. I have just listed them as QY1, QYky, okay, which is not a very good notation. I should have perhaps QY1 and so that is not correct because Y is fixed here. So, indexing has to be only changed, okay. So, put WI equal to intersection of all these, this Q of UI1, UI1, UI2, etc., but take Q of that. I want this one to be an open set. This is a finite intersection. If each of them is open, this will be open. So, what is the guarantee that these are open? These sets are open. Q of that will be open provided Q is an open map. That is why I have put that condition. I want this one to be an open map, open set. This open set definitely contains Y, okay. So, call this as WI, all right. For each point Y, I have constructed an open subset in a very peculiar way, okay. That is very important. Now, you vary the point Y in the capital Y. What do you get? You will get a cover for Y, that Y is compact. Therefore, what you get is a finite sub cover here Y, Y, Y belong to Y. For Y, say Y is contained inside WIJs, J varying from 1 to N. There will be finitely many points. Neighborhoods of those things which I have chosen already, they will cover the whole of Y, okay. On each UI1, there will be N sets sitting above, some KY sets sitting above. So, for each of them you take those sets here, U, Y, J, I. I varying from 1 to KYJ, J itself running over Y1, Y2, YN. So, this is a finite family. For each fixed J, there are finitely many members and J itself runs over a finite. These will cover now the entire of X, okay. That is the whole idea. Take any point X, okay. Q of X must have, must be inside one of these, right. Then Q inverse of that will be covered by one of these elements. That is because of this one. Q inverse of that part will cover by that, okay. So, we have proved one very important result here, namely under subjective open mappings, compactness can be obtained backwards, namely Y is compact and fibres are compact, then X is compact. So, you can cover the reverse way now, okay. This does, this may not hold for lindel ofness, right. Yeah, why? Because you see, you cannot take, you know, you cannot take count, you may have to take countable intersection here because here you may get countable things, right, right. But there are different way you can generalize this one. Think about that. How can I generalize, how far I can generalize, okay. Think about that. There is no time for going into all the details. Directly lindel ofness is not possible here, okay. This statement in general is not, not true also. Not only the proof does not work, in general it is not true. Now another important landmark result, finite product of compact spaces is compact. Converse is easy. Why? Because if product is compact, you can take the projection maps, they are open surjective. Therefore, each factor Xi is compact if the product is compact. It is the converse that we want to attack. But the converse will all come just by this theorem itself inductively. Once I prove it for 2, then you can use inductively, right. So, for 2 what do I do? Look at pi y from x cross y to y which is the projection to the y coordinate. You can, you can check x coordinate also if you want. One of them. This is an open surjective mapping. Y is compact, x is compact. So, x cross, singleton y, all of them are what? They are all homomorphic x. Okay. And they are the fibers here. Pi y inverse of singleton y is x cross singleton y. Okay. So, you can directly apply the theorem to say that x cross y is compact. Now, induct. x cross y cross z will be compact because z is compact and x cross y is compact and so on. So, finite product there is no problem, all right. The theorem is true for infinite products also. And that goes, celebrated theorem of Ticknauf. But for that you will have to wait a little bit. Now, another interesting diversion here suddenly. The following result has a flavor of Cantor's Intersection theorem for metric spaces, complete metric spaces. Here there is no metric, no completion, no deltas and so on. Something funny happens, but you have to put, what you have to put? Start with a compact apological space. Okay. So, let x be a compact apological space. F1 contains F2 contains F3, etc. Sequence of non-empty closed sets. Okay. Non-empty is obviously necessary whatever I am trying to say. They are decreasing sequences they must hear. Okay. And they are closed subsets of the compact set. Then the entire intersection is non-empty. You see in the Cantor's Intersection theorem, finally you had a unique point there. But non-emptiness was very important. So, the same kind of conclusion can be got out of compactness instead of complete metric space and so on. In the complete metric space you needed more stringent conditions. Here is almost less condition. Apply De Morgan law. It is a one line proof that too. Okay. If this is empty, what does it mean? If you take the complement, if the whole space, the complement of the intersection is the union of the complements. What are the complements? They will be increasing, you increasing, you know, subsets each of them open and they cover the whole space X. That means what? By compactness, at some final stage, it must be equal to the whole space. Right? Some Fn will be equal to the whole space because it is a finite cover now. Whatever you take, the biggest one will cover the whole thing. Right? But then what happens? If you go back to the De Morgan law, the corresponding Un is whole space means Fn is empty and that is a contradiction. Fy is a non-empty sorting. So just apply De Morgan law, you get the proof. Okay. So like this, we can go on taking some glimpses of, you know, compact spaces and so on. By the way, there is no such result for Lindelow spaces. So let us do, take, look at metric spaces again and get some more hints for what kind of things we can do with compact spaces. Okay. So that is next time. Thank you.