 Welcome to module 29 of point set topology part 1 course. So, today we will pick up another topic, product of spaces, again within our general topic of producing new topologies, new topological spaces out of the old and so on. Recall that we have already defined the what is called as box topology on at least finite products, right. In fact, we have also done it for infinite products. Now, we want to bring in the most natural way of doing this, whatever the function theoretic approach to the products, not the sub base and basis. Of course, we will also see the sub base and basis here finally. Just like we did some point set understanding, set theoretic understanding of quotient spaces, it is very much important to understand the set theoretic aspect of a product namely infinite products wherein the order implicit order when you take finite products, you write it as x1 cross x2 cross x3. The indexing set has been given a very nice order 1, 2, 3, 4 and so on, right. We should come out of that. It is like baby starts learning bicycle, she is given the support wheels also. So, at that time support wheels are necessary. The order on the indexing set is like support wheels to understand what is going on to begin with, but it is a hindrance for bicyclists in general. So, we should get rid of the support wheels. So, let xj, j inside j, j is indexing set, be a family of topological spaces. Then look at this product space, the Cartesian kind of product space, what a product set first of all. What is it? That is what I want to understand what exactly a set theory what it is. And then on each xj, if you have a topology, then what is the corresponding topology and the product? One of them we have defined, but is that the one or there are some other way of doing it? So, that is the kind of thing they want to understand. And the one which comes out from our pursuit here will be called product polish. Earlier one which we had got was called boxed polish. We shall also compare these two things. First of all, we must understand correctly what is the Cartesian product set, where xjs are some indexed families of sets. I repeat, if j is a finite set, the indexing set, we usually put an order on this indexing set and think of the product as the set of ordered n tuples, x1 x2 xn, where x1 comes from capital x1, x2 comes from capital x2 and so on, xn comes from this. This is what we understand by Cartesian coordinate set s. And this notation can be used x1 cross x2 cross xn. We also know the choice of order on the indexing set is immaterial. x1 cross x2 can be written as x2 cross x1 also for most of the time, unless you are really doing some geometry, something like measure theory and so on. So, there it may be of some value, some importance. Certainly in topological considerations, you do not see any role of this one right now. So, we would like to just be done away with it. So, we shall all together get rid of this order so that this concept can be easily generalized. Recall that the ordered n tuple, now x, I am writing single x for x1 x2 xn. What is it? It is a sequence of length n taking values in union of x1, x2, xn. So, if we get rid of the order, we can still think of this as a function on the set j of n elements which was indexing set there. Taking values inside union of all xj, of course with the additional condition that the jth image of x, x now I am thinking of as a function which is we write it as xj, the jth coordinate that is an element of capital Xj. We see that in this form it immediately generalizes to describe the product when j is an arbitrary set also. The Cartesian coordinate product, we can write it as xj instead of x1 cross x2. What is x1? What is x2 alone? So, the whole product x suffix j, I am writing. This is the product of the family of xj, j inside j. This is defined to be the subset of all functions from indexing set j into the disjoint union of xj's with one extra condition that the jth image, image of x, image of j under x is inside capital Xj for each j this possible. So, we will allow this product notation which is independent of any order on j. This, the left hand side notation is a short notation temporarily for us. Also an element of this product which is a sequence, we will also represent it by xj bracket like this one, like a sequence. So, it is no order here, but we can use this notation, generalized notation. What it means is jth, jth coordinate function, jth value x of j is equal to this xj, this j, that is the meaning. So, they are functions now on the indexing set. The element x is also denoted by this, this sequence following the practice of ordered antipoles or sequences. The assignment x going to xj defines the functions p suffix j from x capital J to x small j which we call jth coordinate projection. Note that pj can also be thought of as the evaluation map. Take an element, take a function here, look its value on the element j, that is the evaluation map. So, these are different ways of looking at the same concepts here. Observe that the product is empty, xj will be empty even if one of the x little j is empty because there is no function into an empty structure. Thus, we shall always assume tacitly that each xj is non-empty while discussing the product. Also, we can further assume that the indexing set is non-empty, otherwise there is nothing to denote. Now, not only that, we will assume quite often that the indexing set is at least two elements set. If there is only one element, it is just xj, there is no product. So, we are not discussing anything. Whatever is stated for product is usually for elements at least, j must have at least two elements, the indexing set and each xj must be non-empty. It is easily verified that a set theoretic function f from y to xj is uniquely determined by the families fjj belong to j of functions fj from y to xj. What is xj? This is fj. This is by definition fj is pi j component f. If you know all these functions, then you know f and conversely. Of particular importance is the case when all the xj's are the same set x, the same set for all j. Nevertheless, while defining the product, we will treat them as if disjoint copies of x. This is just for logical reason, there is nothing more than that. The first one, j, jeth coordinate must be inside xj, that concept is there. Then we can use the notation, this is another notation, x power j, x cross x, you can write it as x square like we are doing it Euclidean spaces, r square r cube and so on. This is r cross r, we do not keep on writing r cross r. So, that notation is available only when all the xj's represent the same set. Then you can write x power j. What is this? This is just set of all functions from x, that is all. So, it will denote xj when the family xj is just x, x, x, x and so on, all of them x. A further simplification in notation in this practice, when cardinality of j is n, instead of writing x power j, you can write x power n, just like we do for r, c, k and so on, k power n. So, that will also, what is this n? It is an element, it is a set with n elements. It is strictly speaking. The natural number n in the class, how many elements are there in that set? It is every integer is a set. How many elements are there? So, that is why this notation is also valid. This is all quite logical, no problem there. A relevant and familiar result in analysis is a function f from y to r power n is continuous if and only if each coordinate function pi i composite f, which we have just denoted by f i also, pi i composite f, each of them is continuous. I am not going to prove this one. This we know, this we have, we have taken for granted. This Rn could be given any topology, usual topology with induced by any of the matrix. There are several matrix, they are given the same usual topology. In particular, we note that pi i, the coordinate functions themselves are continuous. Therefore, in the general setup, when we move from R to X and then from R, from X to X1, X to X, different exercise also. What we want to retain is the coordinate functions must be continuous. Okay. In other words, we have started with Xj set topological spaces. We are hunting for some nice topology on this product space. That topology should satisfy each coordinate function must be continuous. That is the minimum requirement we want to have. Like when you take the quotient set and a quotient function, we wanted the quotient function to be continuous. When you take a subspace, the inclusion map must be continuous. Without that, we do not want to go further. Similarly here, coordinate functions must be all continuous. This can be easily achieved if we take the discrete topology on the domain. If you take discrete topology on the main, everything is continuous. So why bother? But that is more or less useless because it has nothing to do with the topology that we started with on each Xj. So why are you taking these topologies at all? So it has something to do with Xj. So discrete topology will satisfy that. That is like using a atom bomb to kill a fly. So that is not what we want. So we would like to take the smallest topology on Xj such that each Pj is continuous. That is our motivation. Now experience already shows that such a thing is possible. So let us verify, let us go ahead and verify. This is the theorem. There is a unique topology on Xj satisfying the following properties. Each projection map is continuous. Given any topological space Y, any function F from Y to Xj is continuous if and only if the coordinate functions Pj composite F, they are continuous for each j. So this is the claim. So let us see how to do that. Start with all these Xj's with a topology. Let us call that as tau j. Till now we did not need to have a notation for this. Now let tau j be the given topology on Xj. Put S equal to what? Start with Uj inside tau j. Take Pj inverse of Uj. Why I am taking that? Because I want Pj to be continuous. Inverse image of all these open sets must be open so I am putting them here. For all j I have to put. For all j and all Uj inside tau j, take Pj inverse of Uj. That is a collection. Call that collection S. Whatever topology we want to have on this Xj, the product space must contain this S. Therefore take tau S to be the topology. Take the topology tau on Xj generated by S. Remember the my notation tau suffixes for this one. I am not using this one here somehow. It does not matter. This tau is tau S. We claim that this tau has the required property. That first one is that all projection maps are continuous. Well that is how we have managed it here. That is continuous fine. Now let us look at this one. Suppose f from y to Xj is continuous. Then this Pj's are continuous. Just now we verified. The composite will be continuous. So we are only to check whether suppose all these Pj's are continuous then we will f be continuous. The converse part is what we have to worry. Suppose y to Xj is continuous. Then this is continuous follows because Pj's are continuous. Composite is continuous. Now we assume Pj composite f is continuous for all j. To see that f is continuous we can take the sub base S and check that inverse image of f inverse v is open in y. See here there is some topology. The topology has this as base, sub base. Inverse image of sub base if open sets are open is enough to check that something is continuous. You do not have to check continuity on the whole of tau. That is what the claim is we have seen that one. We have used that one several times. So to see f is continuous we must check that f inverse of v is open in y for every y in S. But what is if v could Pj inverse of f uj for some uj in set tau j. So that is an element of v. Each element of S is like that. v could something like this. Pj inverse of vj vj sin tau j. Therefore f inverse of v will be Pj composite f inverse of uj. And this continuous is the assumption. Therefore f inverse of v is open. Very straight forward, right? One step proves there. So it remains to prove the uniqueness of tau. Suppose tau prime is another topology on xj satisfying the same two conditions. By one it follows that tau is contained inside tau prime because this S is contained inside tau prime because tau prime satisfies that Pj's are contained. Once the base, sub base is contained inside the topology is contained inside. So tau is contained inside tau prime. This means that identity function from xj tau prime to xj tau is continuous. So this is the meaning of this tau is contained inside here. Identity map to take an element here. What is in the inverse image and write may be the same element. It must be in tau prime. So tau contain tau prime same thing as identity this way is continuous. Now consider identity from xj tau to xj tau prime the other way around. Okay, if this is also continuous then two topologies will be the same. So when is something is continuous here? Since Pj composite identity is continuous, see I have made the hypothesis tau prime to satisfy two conditions here. Condition one, condition two. Condition one I have used. Condition two I am going to use now. The condition one says that it is large enough. Condition two gives you it is a restriction. It brings down. So that is what is happening here. Okay. So Pj composite f they are all continuous. Where is here? They are here. So to verify something is continuous I have to verify Pj composite identity is continuous. But Pj composite identity Pj itself. Okay. So each Pj f or each j in j property two of tau prime it follows that this is continuous. Therefore tau prime is contained inside tau. Okay. Identity map here is continuous. How to verify this? Use property two for tau prime. Okay. So instead of saying that you see the smallest topology and so on what I have said is I have put another condition here. This condition gives you topology Pj is continuous. The second condition gives you put something restriction on that that continuous functions from y inside xj. So this was from xj to other things namely coordinate from coordinate spaces. This condition is for every y. Any function is continuous if and only if they are coordinate projection coordinate functions are contained. So these two properties define the topology uniquely. You see the theorem. The proof of the theorem gives two descriptions of tau, two different pictures of tau. Half the picture on this side, other other side that will be complete picture. What I said it is the smallest topology of xj on xj actually such that all Pj's are continuous. It is the topology tau s with the base s equal to Pj inverse of uj, uj belonging to tau j. Okay. So this description is more handy for working, for working out things this will help. That will help for conceptual understanding. All right. So next time we shall study product spaces in more detail. Thank you.