 Hello, welcome to NPTEL NOC, an introductory course on point septopology part 2. So, our next chapter is compactifications. The compactness plays a very central role, important role in the study of topological aspects. One always looks out ways of reducing a given problem from a general situation where the given space may not be compact to a situation when the space involved is compact. Compactification is a tool in this direction. We shall study three important versions of it. In this chapter, we begin with the study of Alexander Roth's one-point compactification in full detail, then take up one of the close-related compacts, namely proper maps. After that, we'll study Stonchek Compactification, the third one. I've mentioned three of them. The third one, which is Walman Compactification, that will be later on. It will not be taken in this chapter because that requires some other motions to be developed. So, welcome to module 18. First, we shall discuss some generalities about compactification. Let's expiate a biological space. Let us tentatively have the following definition. A topological space Y may be called compactification of X if the following conditions are satisfied. Compactification space must be compact first of all. So, Y is compact, it's first thing. X must be a subspace of Y. So, the subspace given X is enlarged to another subspace, but enlarged space is compact. So, that's what we want. What is, how far you want to enlarge? Not too large. X must be dense in Y. So, these are the three stipulations you would like to have. Then you can call Y as a compactification of X. At a first glance, the above definition is perfectly all right. However, in order to be able to carry out a comparative study of various compactifications and so on, we shall introduce slightly more elaborate notation. So, technically, we have to be a little more precise. But idea-wise, you have to keep just remembering these three things. For a compactification of these three things, fine. But technically, we will put it slightly differently. So, here is the technical definition. Start with a topological space X. Consider a pair, ordered pair, eta, X twiddle, eta comma Y or eta comma something. So, what are these? Eta, X twiddle, first of all, is a compact topological space. And eta is a map from X to X twiddle, which is an embedding. So, you can use eta to identify X with a subspace of X twiddle. The second condition is, this subspace eta X is dense in X twiddle. So, these two, we are taking care of all the three aspects that we wanted. Namely, this Y, which has been written as X twiddle here now, is compact. Eta is a subspace, eta of X makes X into a subspace of X twiddle. There is a dense embedding. The third one is density, eta existence. Now, you look at two such pairs, eta X twiddle and eta prime X twiddle prime. Suppose, both of them are compact locations as defined as above. We will say they are equivalent if there is a homeomorphism phi from X twiddle to X twiddle prime, such that the subspace eta X and eta prime X remains unchanged. Phi composite eta is equal to eta prime. It is not an arbitrary homeomorphism. The definition such that is very important. So, this is easily seen to be an equivalence relation. Eta X twiddle is automatically equivalent to eta X. Reflexivity is fine. Symmetricness is built in here because if phi is a map from X twiddle to X twiddle prime, which is a homeomorphism, phi inverse will be a homeomorphism, which will also satisfy this property phi inverse of eta prime is equal to eta. Symmetry is fine. Similarly, transitivity is also easy to verify. So, this is an equivalence relation. So, if you look at all the collection of all compactifications of a given space X, you can find this equivalence classes. An equivalence class of such pair is called a compactification. So, this is the final definition. So, what we have brought here is not just some space, not just the embedding, but an equivalence class of them. That is one single compactification. In other words, when you are talking about a topological space, you have the habit of identifying all spaces which are homeomorphic to that space as equal to that. Equal means equivalent. In the same sense, all compactifications of X, X is fixed here, which are equivalent in this one. They will all be called one single compactification. In practice, whenever you are talking about a class, you are always picking up a particular representation of that class, some member in that class. And then you call that itself as the compactification. So, that is the practice that we are following. In the entire class, you pick up any one of them. That is called compactification. So, this much of liberty of language we are taking. But for rigorous definition, we will all stick to this part. So, this is the picture of what is the meaning of equivalence classes. This phi is a homeomorphism, but its identity here, eta component phi is eta prime. So, this is what you have to remember. In practice, we usually take any one particular representative of such a class as a compactification. Though we keep in mind that we are dealing with a representative of any equivalence class. Also, in practice, we ignore specific embedding of X inside X-vital and identify X with eta X. Then, this eta X being subspace X can be thought of as a subspace of X-vital. The above definition can be adopted appropriately depending on the context. For example, you can keep adding extra adjectives for your compactification. Suppose we are dealing with smooth manifolds. There are compactifications which may not be manifolds. But you ignore them. You say, okay, now my X-vital must be also manifold. Suppose you are dealing with only half-door spaces. Then X-vital may not be half-door. In the definition, there is no half-doorness condition. But I may want, don't want to go out of half-doorness. So, I may put extra condition X must be, X-vital must be also half-door. That is the meaning of adopting this definition at various contexts. At different contexts, you can adopt exactly. After that, you don't have to keep on saying that my compactification was also half-doorness. In the beginning of the context, you make it clear. On the set of all compactifications of a game space X, we can introduce a partial order. So, eta X-vital is bigger than or equal to eta prime X-vital prime. I am defining this partial order now. If there exists a surjective mapping X-vital to X-vital prime, again satisfying this property. tau composite eta must be eta prime. You can go back to this picture here. Instead of phi being a 1-1 mapping, you just take it as a surjective mapping. That's all. Rest of the things are same. Then you will call this one bigger than that. So, this must be surjective mapping. So, that is the partial ordering. It's easy to see that X-vital itself is bigger than or equal to A itself. And it is anti-symmetric and it is transitive. Compolt of two surjective maps, surjective and so on. So, this is clearly a partial order. The above revelation is clearly reflexive and transitive. What about anti-symmetry? So, I said it may not be anti-symmetric but I have to be careful here. It is not clear if eta-1 X-1, eta-2 X-2 are two specifications of X. Each one is greater than or equal to other than whether they are equal. That means equivalent or not. See, equality here is equivalent as a class. So, can you prove that? Unless you prove this, you cannot really call it a partial ordering. Something X is greater than or equal to Y and Y is greater than or equal to X. We should have X equal to Y. So, for that reason, you have to be careful here. In general, I don't know what to do but take the special case. Suppose all the spaces are half-dough spaces. Then we are in a fine shape. If tau-1 from X-1 to X-2 and tau-2 from X-2 to X-1 are subjective mappings such that tau-1, eta-1 is eta-2 and tau-2, eta-2 is eta-1. Then it follows that tau-1, tau-2, eta-2 is eta-2. And hence, tau-1, tau-2 is identity map on the dense subset eta-2 X of X-2. Similarly, since X-2 is half-dough and a function, the two functions tau-1, tau-2 and identity map, they are both defined on the whole space and they are equal on a dense subset. They must be equal because the set of points where two continuous functions are equal is a close subset. If the close subset is dense also, then it must be the whole space, that's all. Likewise, it follows that tau-2 composite tau-1 is also identity of X-1. Therefore, tau-1 is a homeomorphism. Tau-1, tau-2 is its inverse. This just means that these two are equivalent. So, if you use the host hardness, then you are in a shape. This is one strong reason why we will restrict ourselves to studying your host or compactification. In our mind, we will keep thinking about host or compactification. But in the definition, we allow ourselves to go out of host hardness. I haven't put any condition of host hardness in the definition. Indeed, we will study the third one which we study. We are going out of host hardness for that. Out of the three examples that I have mentioned. One more remark. Since a compact host or space is a T4 space and hence a Tyknoff space. Tyknoff space is what? 3, T3 and half space. Regular, completely regular and T1. And since any subspace or Tyknoff space is a Tyknoff space, it follows that we will be studying only compactifications of Tyknoff spaces. This is the fallout of, if you just say I am studying only host or space. Suppose you have a host or space and you have a host or compactification. Then the original space, not only just host or space, must be Tyknoff space. If it is not, you won't have a host or compactification. So there is such a strong restriction here if you want to say, okay I don't want anything other than host hardness. Okay, so this is whether we buy it or don't buy it. This is what is there, that's it. So here is a general command. Unless we prove the existence of at least one compactification, all this should be useless. We will be talking in the void, right? It should not happen like that. Of course, when x itself is compact, then you can take x to be x and eta to be identity map, there is no problem. But suppose it is non-compact, then only you want a compactification, right? There seems to be no preferred way to get a compactification of a non-compact space. That is one reason why there are several solutions to this problem. Perhaps a method which may immediately occur to one's mind is the so-called Sx, the shear-pinsification of x. However, even if x is host or Sx may fail to be so, all the time Sx is always non-hospital, unless x is empty and Sx has a significant point. So if you want to retain house durveness, taking Sx is useless, right? Start with any house durveness space, you may not get a house durveness competition at all, okay? However, we now know that there are compactifications of non-compact spaces, okay? Namely, you can take Sx if you are ready to go out of house durveness. At least we are not working in a void, there are compactifications, okay? From now onwards in this section, we should assume that x is non-compact. There is no point in discussing compactifications of a compact space, okay? So I may not mention x is non-compact again and again. For each n belonging to n, namely natural number, by an n point compactification of x, we mean a compactification eta, y, where y minus eta x is precisely n point. That means I have added exactly n points to the original space, x, okay? That is the meaning of it. For example, it is easy to check that 0, 1, the closed interval, and this eta is inclusion map, okay? From closed interval 1, 0 to open 1. Take the inclusion map into the closed interval. What you have done, you have put one extra point. Or you can start with an open interval 0, 1, and add two points, 1, 0 here, and other end, 1. So you get this space is a one point compactification of this one and two point compactification of this one. The inclusion maps are depending upon whether you start from 0, 1, open, both 0, 1, open. 0, open, 1, open, okay? So you get a one point compactification as well as two point compactification. So this is an illustration, yes, that's all. It's very easy to see. I have a question here, open question for you. Think about it. Even if you don't answer this, okay? Can you think of a three point compactification of 0, 1, open, control 0, 1, which is Hausdorff? See, both these examples were Hausdorff space, okay? So that is a question. So think about it, that's all. Okay? So some more general remarks. Compactifications are always studied with some extra specifications depending upon the kind of problem that we are interested in. It is not possible to discuss all of them, certainly in an introductory course. So in this section, we shall study two such examples. Later on, we shall study one more. These three being the most important ones in our mind. Okay? So I have already told you, namely, Alexander's one point compactification and the Sto-Czech compactification. These two things we will study in this section. All right? So let me begin with Alexander's compactification. Let X tau be any topological space. X tau is a disjoint union of X with one extra point which I am going to denote by infinity. This is modeled on what you call the extended complex plane. You must be using this notation where X is the complex plane C and that is infinity. There is no algebra or anything here, okay? Neither even analysis is just points of topology. If you remember that. Let tau star be the family of all subsets A of X star such that if A inside X, then A must be tau. All subsets of X which are inside tau, they are allowed inside tau star. That is the first condition. Otherwise, what are they? They must be containing infinity, right? So X star minus A, when you throw away A from X star, what you get? You will get a subspace of X, right? So that must be closed and compact subset of X, okay? So this see the condition on tau star. Now this tau star becomes a topology which is compact, topology on X star. Whenever X is not compact, this is the standing assumption I have, but here this construction I could have done it for even when X is compact. If it is of any use, you can use it. Otherwise you can throw it away. But for logical reasons, I could have done this one even for X. It's a compact set. That is why I am telling whenever X is not compact, that is our central theme here. Then what happens? Eta X star is a compactification of X where eta from X to X star is inclusion map. See, this is just a set theoretically. It's just X union infinity. So that is an inclusion map. Further, X star is half star if it only X is half star and locally compact. Once again, we are hitting the notion of locally compactness here in Alexander's Compatification. Alexander's Compatification is able to achieve house dwarfness provided we start with house dwarf and locally compact space. So this is the conclusion. So let us do this one by one. One or two things we have to verify here. First thing is tau star is a topology. What you have to do? Empty set is there because empty set is a subset of X. The whole space is there. Why? Because if you throw away the whole space, it's empty set. That is the condition here. That is compact and a compact set of X. So that is easy to see. If two subsets are there already in tau, their intersection will be also in tau. So it's also there. If one of them is inside A, another one is contained in infinity, their intersection will again be at tau. So union, arbitrary union of open sets is there because once one of them is closed and compact subset, the entire arbitrary infinity, that will be subsets of that one. So it's easy to verify or re-verify it already. That tau star is a topology. This topology, when you restrict it to X, becomes precisely tau. So the inclusion map is a homeomorphism. Inclusion map X to X, X to X infinity is a homeomorphism. Because every open subset here becomes open subset and vice versa under the inclusion map. So if Ui is an open cover for X star, I have to verify that X star is now compact. Let us say infinity belongs to U1. It has to be in one of the open sets. Put F equal to X star minus U1. U1 being open subset containing an infinity, F must be compact subset of X star, closed and compact subset of X star. And Ui or the Ui intersection X, I not equal to 1, that will be a cover for X. You don't need I equal to 1 because that part contains infinity. Rest of them have to cover this one. Therefore, we will get a finite cover U1, U2, Un, such that F is contained inside U2 intersection F, etc., Un intersection F. Then it's clear that X star will be contained inside U1, U2 and Un. You have to just come back to X by using one subset, which contains one open subset, which contains infinity. So X star is compact. Now come to the hypothesis, if X is not compact, then it follows that every neighborhood of infinity should be intersect X. If X were compact, infinity itself would have been open subset, singleton infinity, because complement is compact and closed. X itself is compact and X is closed inside X. So that is not the case. Singleton infinity complement will be X and it's not compact. Therefore, it is not, singleton infinity is not open. So every open subset must contain some point of X, that's all. Which just means that the closure of X is the whole of X star or existence. This shows that eta component X star is a compactification in our definition. It may be worth to note that X itself is an open subspace of X star. Singleton infinity is closed, why? Because it's complement is the whole of space X and that belongs to tau. Therefore, it is in tau star. So X itself, eta X if you want to say, is an open subset of tau star, X star. Open and dense, this much we have seen. Finally, we want to show that X star is off star, if it only exists locally compact in the off star. If X star is off star, being a subspace, X will be off star. That is easy. Why it is locally compact? Let X belong to X and Y belong to X star with any two points. So I am talking about now, why X star is off star. If you take two points of X, distinct points of X, you can separate them inside X itself. Therefore, they are separated inside X star also. So now, suppose one of the point is Y, X belongs to X, Y belong to X star with any two points. If Y is not infinity, take distinct open subsets U and V, such that X and Y are inside V. Both of them are open in X star also, those are done. If Y is infinity, that is important place, using local compactness of X. We may assume that the closure of U in X is compact. You can start with an open subset around U, such that U bar is compact. Then X star minus U bar is a neighborhood of infinity in X star, which is distant from U. Therefore, X star is off star. So I am proving here X star off star, using local compactness and off star sense of X. Converse part I come now. Suppose X star is off star, then being a subspace, X is also off star. Moreover, for X belonging to X, if U and Z are distinct open subsets of X star, such that X is in U and infinity is Z, then by the very definition U is inside X is open. X star minus Z is a closed and compact subset of X. Therefore, U is contained in X star minus Z because they are disjoint. X star, U and Z are determined, U is contained in their complement. It follows that U bar is compact because U bar is also a subset of, is compact. Hence X is locally compact. In Hofdorff's space, we have proved that there are several equivalent conditions of local compact. For each point, if you produce a neighborhood such that closure is compact, then it is locally compact. So what we have got is that Alexander's compactification is a one-point compactification. It will be Hofdorff if we start with a Hofdorff and locally compact space. And that is the most important special case that we are going to study further. So I make this definition. The above compactification of X is called Alexander's compactification. Alexander of compactification. Note that Alexander of compactification is a one-point compactification. In the literature, it is common practice to refer to it as the one-point compactification of X, especially when X is locally compact and Hofdorff's space. See, there are many one-point compactifications. The compact equation that we have constructed, namely Alexander of, is a special one. But in the literature, it is the one-point compactification. Whenever we start with locally compact of X, people always refer to this one as one-point compactification. So we may also do that. If it is not Alexander, we will specifically mention it, that's all. So we may sometimes follow this common brand name. Is that clear? Okay, so let us stop here. We shall continue this study of this one and bring the new concept of properness, etc. Next time. Thank you.