 Welcome to model 34 of point set topology part 1. Last time we introduced the notion of connectedness, motivating it from the notion of path connectedness and then relating it to intermediate value theorem, least upper bound property etc. So, we will continue now the study of connectedness. So, here is a simple theorem which says that if you have a continuous function from one space to another space, it will take a path connected space as well as a path connected space whatever to the corresponding the image will be connected or path connected. So, it is actually two different statements here. By taking a subset Z which is say connected or path connected and restricting the function to that and to the domain FZ on the co-domain FZ, we can as well assume that F itself is surjective and make the hypothesis that X itself is connected or path connected. So, let us first look at when X is connected. I want to show that Y is connected. So, take a separation of Y, Y equal to a separately, it pulls back to a separation of X via F. So, all that you have to do is F inverse of A, F inverse of B that will be separation of X, pure synthetic thing. Continuity of F is used only to see that these two sets here F inverse of A and F inverse of B are closed, disjointness, union is the whole of X is all pure synthetic. So, that proves that image of a connected space is connected. Now, let us assume X is path connected and we want to show that Y is path connected. So, remember now I have assumed F is surjective. So, given any two points say here A and B, sorry X and Y or X1 and Y and whatever, they are F of A and F of B for some points A and B instead of X. But A and B inside X can be joined by a path omega or gamma. Gamma is a path in X joined A and B. Then F composite gamma will be a path joining F A and F B. That shows that Y is path connected. So, we had defined long back that a property of topological spaces is said to be topological property or a topological invariant if the following holds. Whenever P is true for a space, it must be true for all spaces Y which are homeomorphic to X, which are homeomorphic to X. So, that was the definition 1.59. I am just recalling it. So, because image of a connected set is connected automatically under a homeomorphism image of a connected set will be connected because homeomorphisms are on to. So, homeomorphism preserves connectivity. Similarly, it preserves path connectivity. So, these two are topological invariants. Because of this property, what happens is whenever you are studying some connected spaces and so on, you can actually assume the whole space is connected, concentrate on a connected part of it and then discuss the whole thing. Because once you have that continuous functions from there will be always inside another connected set. After all, we are all the time studying continuous functions. So, lot of such discussions can be isolated just around a connected set. So, this leads us to the notion of what is called as connected components. Suppose Z is a connected subset of X and X has a separation, then either Z is inside A or it is inside B. So, this is the property of a connected set. Otherwise, what happens? You look at the, you know, you have to just take the restriction to Z, Z contained in ZX, Z intersection A, Z intersection B. That is separation of Z. Only thing is you do not know whether each of these subsets are non-empty. Z intersection A will be non-empty if Z is, you know, not contained inside B and vice versa. If not contained inside A, then this will be non-empty. So, if both of them happens, it should be non-empty and that will be contradiction that Z is connected. So, one of them must be empty which means Z is contained inside the other, either A or B. Okay. Therefore, what happens is, okay, let us, let us understand one more thing about this connectivity. Suppose you have X as a union of two subspaces which are both connected and the intersection is non-empty, then X itself is connected. Later on, we will generalize this one. Okay. So, this is just a trick using the previous lemma. Suppose this union has separation. Okay. I have assumed that X1 intersection X2 is non-empty. So, take X to be a point in the intersection. Then X being a single point, it must be either inside A or B. So, you can assume it is inside A by changing the notation if you want. But then, from above lemma, it follows that, you see, X1 contains little x, right? And that is connected. X1 is connected. Okay. So, Xi must be contained inside completely inside A. It is contained inside A or B is the previous lemma. But since already X is inside A, it is a common point with X, X1. So, it follows that X1 is inside A. The same argument for X2 also. So, both Xi's are inside A. That means B is empty. So, that is a contradiction. This just means that there is no separation of X. And hence, X is connected. So, you can see that already this is very easy to generalize this for any family X1, X2, Xn, not necessarily even finite. Okay. Here, I have not used any indexing set as just 2 here. This can be just arbitrary indexing set. It will be applying for all. That is just an observation you can make once you have understood for X1 and X2. So, now we make a definition motivated by especially this lemma. Okay. And then this corollary theorem here. Okay. Let P be a statement about subsets of a given set X. This is a general definition first of all just to tell you what is the meaning of this maximal. We say a subset of X is maximal with respect to P. First of all, it must satisfy P that property whatever. So, A satisfies P that is necessary. Then among us all those who satisfy you know this P say suppose B also another member which satisfies P. Okay. Then A cannot be properly contained inside B. B must be you know A must be the biggest or A must be the maximal that is the mean property. So, if B also satisfies A must be equal to B. A is contained inside B and B is also satisfies this A must be equal to P. This just means that if I take slightly larger subset strictly larger subset then it will not satisfy P. Okay. So, this property P must be for subsets of a topological space X that is the whole idea because it is a topological proper 10-way road. So, whereas this definition is for all this is just a synthetic definition. There is no topology here. Now we are going to make it a topological property especially take P to be property of being connected. In other words look at all subsets of X collect all of them which are connected into a subspace. So, that will satisfy P. Right. So, P is a subset of the power set of X. All elements are connected to begin with. Now you take a maximal element inside this collection. Okay. That means anything every element here is connected first of all and anything slightly bigger than A will not be connected. Okay. That is the meaning of maximal element. Okay. So, we can make that as our definition now. Let us see. Let X be a topological space. Then every point X belong to X is contained in a maximal connected subset. Thus X is the disjoint union of its maximal connected subsets. If I prove the first part then the second part is purely set theory because every point is in a maximal set. The union of maximal subsets will be the whole of space. So, we want to prove that each point is contained in a maximal connected subset. To begin with singleton sets are connected. Right. So, they are the members of this set of all connected subsets of X. All right. I want to show that it is in a maximal set. Everything every X must be in a maximal set. That is what I want to show. Okay. So, now use this property to become, you know, to make, to grow this set point, single point into a maximal set. That is what we want to show. This property we keep using again and again. So, here is the way how to do that. Define a relation. I am saying here is a way. There is not just, this is the only way. There are many ways. For example, you can use John's lemma and so on. I do not want to, I want to do it very elementary way. Define a relation R on X as follows. X is related to Y if and only if there exists a connected subset T inside X to which both X and Y belong. Both X and Y belong to T. Singleton sets are reflexive. So, I do not have to take anything. I can take T as singleton X. Then X is related to X. If X is related to Y, XY's are both inside T. So, Y is also related to X. Transitivity is previous, is precisely what is needed and that is previous theorem. If T1 is containing X and Y and T2 is containing Y and Z, then what happens? T1 union T2 will be connected because they both of them have common point Y. T1 and T2 are connected. They have common point. So, union is connected is a previous theorem. So, I have found a connected set which contains X and Z. So, transitivity follows from the previous theorem. Therefore, this R is an equivalence relation. As soon as you have equivalence relation, you have a partition. Then this is the second part was precisely a partition. What is partition here? X is the disjoint union of maximal connected subsets. So, each member of the partition is a maximal connected subset. So, that is the part directly improving here. That will give you that every point is inside a maximal connected subset. So, these equivalence classes are I have to show connected and they cannot be bigger. If you put any more point, it will be disconnected. That is a maximality. To show that it is connected, take an equivalence class. Suppose there is a partition, pick up an element A in A and B in B. But they are in the same class which means A is related to B and hence there is a common connected such T which contain both A and B. But T be connected, it should be either in A or in B. That is a contradiction because one point is inside A, another point is inside B. So, T is first of all a connected subset and C is the equivalence class of all of them. So, T, all the points of T are inside C first of all because T is connected. All elements of T are in one single class and A and B are in C. Therefore, the whole T is inside C. So, but this partition for T intersection A, T intersection B will be a partition or T that is not possible. So, the different ways of arguing, I can just show that T, previous term already T has to be contained inside A and B. So, that is not possible. Therefore, C is connected and there is no partition like that, there is no separation like this. Therefore, each equivalence class is connected. The equivalence classes are already partitioned. So, anything bigger than that cannot be connected. But if it is bigger than that then it is not a class, it is a bigger class. So, these are actually maximal connected sets. So, it is better to give a name for these maximal connected sets after all. All the time otherwise you will have to keep on saying maximal connected, maximal connected. So, such things are called connected components and quite often when you are discussing connectivity, you will just say component. That is the whole idea of naming this. I mean this definition is just for the name sake. Precisely, I mean literally it is for name sake. Let Z contained inside X be a connected subset. Then the closure is connected. Now, why we are having such a thing? You see, now we want to understand what happens to connected components. They are already giving you partition of the whole space. Now, this theorem says that take a connected component. Closure is automatically larger, but it cannot be larger. It has to be equal because closure is also connected component, connected subset. Being Z being connected component, anything bigger cannot be connected. Therefore, there must be equality. So, this theorem tells you immediately that connected components are closed inside the original space X. So, let us prove that Z bar is connected. Assume Z bar as a partition, A, B, separation. But then Z is connected. So, Z is inside A or inside B. Remember, if A is closed, Z is contained inside A, then Z bar is contained inside A. This was our old result about closures. Therefore, Z bar is either inside A or inside B. So, there is no such separation. So, you see several of these results we have been reducing by contradiction. Why? Because the definition of connectivity itself is in that form. That is what I meant by it is some kind of a negative definition. Path connected components are also defined the same fashion. What is it? Maximum path connected subsets of a given topological space. So, you can also say that just the equivalence relation is that if X is related to Y, there is a path from X to Y. No need to have their path connected component. A path, image of a path is already a path connected space. So, this is slightly easier to digest equivalence relation. In fact, usually whatever happens to path connected spaces, you try to copy it in connected and if it works, it works. So, that is the way that perhaps this theorem has been used here. This definition has been done here. So, path connected components also give you partition of the whole space. One has to be very careful when you keep saying it is same thing, same thing, same thing. It is lot of close relations but they are after all different notions. So, somewhere they will be different. So, that is what you have to be careful about. That X be a topological space and X belong to X be any point. Then the set of all points of X which can be joined to X in X is the unique path connected subset of X containing that point. So, this is another description of path connected components. Start at any point in a space, look at all those points which can be joined to that. So, that has to be obviously that is path connected that has to be the component. If there is another point that can be also joined, it is already there. That is all, very easy to look at this way. So, that is no need to write down formally the proofs of this one. This is where I want to caution you that path connected component need not be closed. In other words, if Z is path connected, Z bar may not be closed. If that was the case, then components would have been closed. So, we will see an example little later. Secondly, if we have homeomorphism from X to Y, then look at the path connected components of X. They will go in one-one fashion to path connected components of Y. Exactly, the components of X, connected components of X, they will go in one-one fashion to connected components of Y. In general, path connected components are connected also that we have seen. But number of connected components may be larger than the number of path connected components. But this correspondence is true for both of them. Therefore, what happens is, suppose you want to analyze a homeomorphism at arbitrary X and Y and something. You can do that by component wise, restrict F to one component here. When you go to the image, it will be another component there. Therefore, right in the beginning, you can assume that both X and Y are connected by restricting the whole thing to a connected component. So, this is how I already told this one, but I have repeated it now. So, next time, we will make sure that you will be able to see a counter example also for this. In any case, we have a lot of work to do about connectivity. Simultaneously, whenever such things are true for path connectivity, we will keep informing you or we will keep pointing out to you. Let us stop. So, until next time, let us stop here.