 Hello, welcome to NPTEL, NOC and introductory course on points of topology part 2. So, today we will do local compactness. Both compactness and lindelofness can be termed as properties which are global. Global topological properties have certain local versions also. Often it may turn out that global version of the property may not imply local version of property. For example, we are still connected space and also locally connected space. Connectedness need not imply local connectedness also we have seen. On the other hand, you know the first countability can be thought of as the local version of second countability, but second countability implies first countability. So, do not jump to the conclusion either way that is all I want to say. In general, one should not expect a global version of the property to imply local version of the same. Compactness is another important property which admits a local version. In the study of topological vector spaces, we have already made an add-off introduction to this concept in part 1. Let us now consider it in full generality. So, whatever we did in the study of vector spaces, topological vector spaces, it was an add-off thing that will be recovered automatically. We are not abandoning that. Let us begin with a definition. Start with a topological space, we say x is compact, sorry locally compact. If for every point x inside x and every open subset u in x such that x is inside u, there exists an open set v such that x belongs to v contained in v bar contained in u and v bar is compact. So, at each point and each open set, this should happen, it just means the following thing. The set of you know compact v bar can be thought of as compact neighborhood, why this is a neighborhood? Because it contains an open subset contained in here and we have made it compact, ok. So, compact and closed neighborhood, v bar is closed also, compact and closed neighborhood of a point, ok, they form a fundamental system of neighborhood, it just means that for each point each open subset containing there is a smaller one contained, that is the meaning of fundamental system of neighborhood. It is another way of saying that x is locally compact, ok. It should happen at all the points, so localness can be sort of x is locally compact at x you can say like continuity at a point and then continuity on the whole space means continuity at each of this point, it is similar to that, ok. So, that is the definition of local compactness. Now, I want to tell you that there are other definitions, slightly weaker and more and more weaker and so on, ok. So, so let us be careful about this definition, ok. So, I repeat a space is locally compact if and only if each of its point that is a neighborhood system consisting of compact neighborhoods, in fact compact and closed neighborhood. Note that this definition of local compactness, ok, regularity comes automatically, see just x belonging to you, there is a such a neighborhood without even compactness is regularity. So, perhaps my definition is too strong, ok. It is not my definition, it is there in the literature. Out of the several definitions, I adopted for this definition of local compactness, ok. So, I want to tell you that some authors prefer to have a weaker condition of locally compactness, ok. We give two such instances which are very much common, there are many others I cannot go on dealing with all of them. These two are quite important also, so I would like to incorporate them here, ok. In a theorem, start with the tabloid space, consider the three conditions here, three different statements. Final conclusion is that the third one implies the second one implies the first one. If x is regular or half star, then all of them are equivalent, one implies three, three implies two implies one, so one implies three they are all equivalent under regular or half starness, ok. So, what are these three? The first one is very simple namely for every point x there exists a compact neighborhood which you may write kx of x, that is all. The second thing is slightly more elaborate for each point x inside u where u is open, there is a compact neighborhood kx of x such that kx is contained inside u. So, this you can reformulate like saying that x belongs to u, sorry x belongs to v, v contained inside kx, kx contained inside u and kx is compact. The third one is our definition of local compactness. The difference between two and three is that x belongs to v, v contained inside v bar, contained inside u and v bar is compact I was saying. Here some compact set comes, v bar itself may not be compact, ok, some compactness. So, that is the difference between two and three you can state. However, very easily three will automatically implies two, ok. Because I can take v bar in place of kx, three implies two, two implies one automatically for each point it happens you can take u to be the whole of x then there is a compact neighborhood over. So, two, three implies two implies one is very easy. So, what I want to show you is that assume house dorseness or regularity then this one which is the weakest here implies three namely our definition of local compact, ok. So, these three can go hand in hand quite often the underlying topological space is regular or house dors. There are two schools of anthropologies, some of them take regularity all the time, some of them take house dors all the time, very few people who do not deal with either regularity or house dorseness. So, in that case all these three definitions are equivalent. This is very, very important to you know important to notice. So, that is why I have put it as a theorem here, ok. So, let us give the proof. The proof of A, A part is what that is what I have told you know the three implies two implies one. Now, let us come to proof of B under regularity take x belonging to you contain inside x where u is open by the hypothesis what we have? We have an open set vx and a compact set kx such that x belongs to vx contained inside kx. So, this is the hypothesis one. Now, I use regularity, ok. There exists an open set w such that x is inside w contains a w bar contained inside this starting with u and vx. u is open subset around x, v is also opens vx is also open subset. So, u intersection v is an open subset containing x. So, regularity gives you a w contained w bar in that way, ok. Now, w bar obviously is inside vx and vx is contained inside kx, ok. w bar is a closed subset of kx but kx is compact therefore, w bar is compact and that is why I wanted x contained inside w contained inside w bar compact contained inside u. So, this implies three, ok. So, we are done with one part of part B. The second one is regular, instead of regularity, Hausdorffness, ok. Now, suppose x is Hausdorff. Start with a point x, ok. And ux be an open set such that x belongs to vx contained just like in the previous case. Being a subspace of a Hausdorff space, kx is also Hausdorff. Now, kx is compact and Hausdorff. Therefore, our previous proposition there is its regular. So, now kx is regular, ok. Therefore, as seen above, kx is locally compact. So, I am using the earlier part here, ok, so that I do not have to work again too much. So, now I have all instead of x, I can assume kx or I can assume x itself is regular. See, it is x itself is locally compact. Now, suppose u is any open set in x, the set x is inside u. So, I want to show that the whole thing is working inside x itself, ok. Then u intersection ux is open inside kx and it is a neighborhood of x. Hence, by local compactness of kx, we get an open set wx in kx such that x is in wx contained in the closure of wx contained in u intersection ux contained in kx, ok. So, here I have used the local compactness of kx and u intersection ux in neighborhood kx, ok. I am writing this closure here just to indicate that now everything I am taking inside kx is important. If I can go directly to x, then the proof is over and that is precisely what I want to indicate now, ok. The closure of wx is compact, ok. So, that is the conclusion because now we have used the second part here named kx is regular, locally compact. Now, the closure of wx denotes the closure of this wx inside kx. But wx is open in ux intersection v. This is open here and u intersection v is open in x itself. Therefore, wx is open in x itself. Now, kx is a closed subset of x, ok because locally compact or star space, ok. I have compact or star space, right. It follows that closure of wx which is equal to now vx bar itself, ok. Because kx is closed, this will be closed. If you take closure of this in x, it will be same thing as wx bar. It will not give you anything else. The closure of wx in x itself, ok. So, this is the fact which allows us to say instead of writing this new symbol here, I could have taken wx bar. That is justification, ok. You could once you have, once you have observed that kx itself is, you know, locally compact, you may say, ok, go ahead. So, this is justification why you can go ahead. Some justification is needed because closures are being taken in different subspaces. Take an arbitrary set A subset of y, subset of x. The closure of A in y and closure of A inside x, they can be quite different, right. So, that is why I needed to justify this step. That is all. So, here is an example. An indiscreet space with more than one point is an easy example of a locally compact space which is not housed off, ok. An indiscreet space is actually compact also. Because what are the open sets there? The only open set is an empty set and a whole space. So, but these are non-hospital space. So, you have to be careful there. Any infinite set with co-finite topology satisfies property 2, but it is not locally compact. Why? Because the closure of every non-empty open set in the co-finite topology in the whole space. Every non-empty open set is dense. So, closure of the whole space. So, x belongs to you, you say wx I can choose. But moment I take wx bar, it will not be contained in my smaller open set. It will be the whole of space. I can start with a proper open set and take a point inside that. I cannot get a compact set sitting inside, ok. So, any infinite set with co-finite topology satisfies 2, but does not become locally compact in our definition. So, under Hofstorzness they are equal, under regularity they are equal, but this co-finite topology is neither. That is the problem. So, there are such interesting examples wherein property 1, property 2, property 3 may be all different. Property 1 is true for all indiscrete spaces, ok. An open quotient map preserves 1, ok, because you need all that you need is a compact neighborhood, ok. A quotient map preserves open sets to open sets, compact to compact. Therefore, you will get a compact neighborhood of the quotient points in the compact topology, ok. Similarly, it will preserve 2 also. But not necessarily the third one, namely our compactness, ok. So, in part 1, we have studied an example of an infinite cyclic group acting on R2 minus 00. Namely, the action was an integer comma xy going to 2 power n times x, y divided by 2 power n times x, 2, y divided by 2 power n. So, the x coordinate is multiplied by 2 power n, y coordinate is divided by 2 power n. This is a topological action of a discrete group result, ok. So, this action was used for several counter examples in part 1. This space is a T1 space, the quotient space, but it is not hostile. The image of 1 comma 0 and 0 comma 1 in the quotient space cannot be separated by open set. That is what we have seen anyway. It is not hostile. If it were locally compact in our definition, then it would imply that it is regular. Regular T1 will be hostile, but we have seen it is not hostile. Therefore, this quotient space is not locally compact. A z action automatically gives you open quotient. So, this is an example of a locally compact space. R2 is locally compact. R2 minus 0, it is also locally compact, ok. This we know. This is very easy to see, ok. Given any point, you can take a closed ball around that inside any open set. Given any open set, right. The closed balls in R2 are compact. That is what we have seen. So, R2 minus 0, 0 is locally compact, but this quotient space is not locally compact in our definition, ok. So, here is a remark. A compact set need not be locally compact. In the definition 1, if you take it is, because each point you can take the whole space as a neighborhood, which is compact, ok. In our definition, it doesn't work. All you have to do is to start with a space while it is not locally compact, like I have given you many examples here already, ok. There are many. Then you can add one more point, namely this is adding one more point. It is, I have introduced this one in part 1 very meticulously, because I want to keep using this one. This is called shear-pin skippification, ok. So, this shear-pin skippification has the property, it adds one more point. But what are the neighborhoods of this point? The whole spaces, that's all. All other open subsets in Y, they are there. That's a topology. Obviously, any point which contains this extra point must be the whole space. Therefore, there is a finite cover of the whole that opens up. We can take it as X. So, this is automatically compact no matter what Y is. Y is the subspace of this, but I started it not locally compact space. At all those points, local compactness will fail, because Y itself is an open subset of this one. If you take point X inside Y, ok, you know that there are such points where in X, say little Y belongs to Y and there is no locally compact, there is no compact neighborhood and so on, ok. So, this way you can produce, you can make a locally compact space, not locally compact space, convert it into a compact space by just putting an extra point, ok. The local compactness is not hereditary either. Recall hereditary means whenever something is locally compact, all subspaces must be locally compact. So, that will not be the case here. Again, very easy to see such things. So, what is interesting case, ok. So, here, sorry, here is an interesting example first of all before going to further. Take the very, you know, interesting case namely rational numbers inside R, very popular example, ok. I want to say that R is locally compact, that we know, because every open set around a point, you can take a closed, closed neighborhood around that, that will be compact, right. So, we want to show that Q is not locally compact. What is the meaning of that? Given a point and a neighborhood, ok, I must produce a compact neighborhood. If this fails at at least one point, then it is not locally compact. Here we are going to show that it fails every point, every point and every neighborhood, ok. You can choose any point and any neighborhood of that inside Q. What is a, you know, a neighborhood? A neighborhood must be a neighborhood inside R intersection with Q. Only rational numbers inside an open interval. Take such a thing, take a point for example, 0, ok. Take an open interval around 0 and then take all the rational points in that. That will be the subspace, open subspace of Q, ok. I want to say that there is no compact neighborhood of that one, ok. So, what we want to check is the following. Start with any a less than b so that a b is an inter open interval. Take the intersection with Q, ok. If you look at the end points or irrational numbers, the closed interval a b intersection with Q is the same thing as open interval a b intersection with Q, ok. This will not be compact, ok. So, this is enough to tell you that you cannot have any compact subsets of any interval inside Q, alright. So, Q is not locally compact. This is easy to see that a b is not compact because a and b are irrational number. I can take a sequence of closed subsets here, closed intervals here, ok. Which tend to a b, union will be a b, countable union of these things. So, that will cover for this one. No finite sub cover will be there, ok. However, the saving grace is that local compactness is weekly hereditary. Weekly hereditary by which we mean a closed subset will have that property or sometimes open subsets will have that property. So, there are two versions of weekly hereditaryness. In both cases, this will work here. Every open subsets of every and every closed subsets of a locally compact space is locally compact. So, then you can combine them also. Suppose you start with a locally compact set, compact space and then take an open subset that is locally compact. Now, you take a closed subset of that that will locally compact. So, you can play this game. So, what it is called as, you know, this is called locally closed subsets. Very nice terminology. It means just close inside an open set or open inside a closed set. So, such subsets will be also locally compact, ok. So, I want to recall a few things here. In part 1, we define a topological vector space v to be locally compact just by the small condition. Namely, 0 has a neighborhood o with o bar compact. So, this was even weaker than our definition 1, right. Just for 0, if it satisfies, then we are done. But in the case of topological vector space is something happens at 0, ok. It will happen at all the points by translation. If there is a open subset around 0, translate it at any other vector. So, that will be an open subset containing that vector. This closure will be translation of the closure, translations are homeomorphisms. Therefore, it will work for all the points. That is not only the condition. It actually finally, it is so strong that we have been able to prove that under this condition, the topological vector space is actually linearly isomorphic to some finite dimension Euclidean space. So, this was one of the important result that we had proved in part 1. Once it is R n, it is locally compact in our sense also, in the strongest sense, ok. So, that is why it was enough to assume such weak condition there for local compactness because we are working in a topological vector space, ok. You will meet some other interesting examples of non-locally compact spaces in algebraic topology wherein you have to take unions of, you know, infinitely many spaces and identify them. So, I will not be able to go through that here. Lindelofness is also one property which we can consider local versions. Exactly the way we have done above, namely in three different ways if you like, right, ok. Just like for local compactness. This has been studied by some people, some authors. There are papers you can Google, you can search through that, ok. However, local Lindelofness doesn't seem to have many customers. So, we will not discuss it any further, ok. So, that is enough for today. There will be many more things about local compactness. So, we will discuss it next time. Thank you.