 Welcome to module 31. Today we will sum up the entire chapter of constructions of various topologies and some superficial study of these things so far as an introductory chapter. So all these constructions namely subspace, unions, quotient, product, they can all be put into two types of constructions. One will be called as induced topologies, other one will be called co-induced topologies. So that is the topic for today. So let us first consider induced topology. Take a situation where we have a set X and a collection of topological spaces Y i and functions F i from X to Y i. So this is the situation we are facing with. We want to somehow relate the set X with the topological spaces Y i through these functions F i in a topological way. Relating means what? Already there is a function you can say okay, go ahead. No, you want to bring in some topological aspect here. So in order to bring out the topological geometrical aspect of these relations, it becomes necessary to put a topology on X so that the functions F i's are first of all continuous functions. You can say that it can be easily done because you can always take the discrete topology on X, then all the functions are continuous. So go ahead you may say. No sir, that won't do because the discrete topology is too good in some sense. Therefore it's useless also because no matter what X i's, what Y i's are, what kind of topological space they are and what F i's are, the discrete topology is going to guarantee them that all these things are continuous. So it does not relate what's happening in Y i at all. So we want to have some, the topology to do something with these Y i's and something with the F i's. So that is one thing why we reject taking discrete topology. Second, it is too good. It is too much to be expected. It is not economical to put so many open sets. Okay. So that leads us to the following definition. We want to do it economically. That is the keyword here. Given a family of functions, you want them to be continuous. So this definition takes a unique smallest topology tau on X such that all the F i's are continuous. We have seen this. So this topology tau is called the topology on X induced by the collection F i. The induced topology, remember, this is the definition. What is the smallest topology? What is it? It has the sub-base consisting of all F i inverse of U i's where U i's range over tau i. Take any element of tau i, say U i. Take F i inverse of U i. Put all of them together. Okay. So this is the topology. Union of all F i inverse of U where U belongs to tau i. That is a sub-base. Okay. Often it is also called weak topology with respect to the family F i. This name weak topology is justified for the following reason. Suppose you start with a topological space X tau and a family of continuous functions right in the beginning. Then the induced topology on X from this one will be obviously weaker because by definition smaller is the smallest one. So it will be smaller than this tau. Anything which satisfies that all of them are continuous will be containing the smallest topology. So that is why it is called weak topology. Weak means weaker means coarser. Smaller that is all. So let us go back and examine whatever you have done already. So it is not a new thing almost. I told you it is all summing up this chapter. Consider a special case when Y is a topological space and X to Y is just a subset. X is subset of Y. Now you take F equal to eta equal to the inclusion map. Okay. X has no topology. Now what we have done? We have put the subspace topology here. But just do this one namely take the inclusion map and put the induced topology whatever we have defined just now. Then what is this topology? I have to take an open set here. Take eta inverse of that open set which is nothing but open set U intersection with X. Okay. Therefore this topology is easily seen to be induced by this nothing but the subspace topology. Can you intersect it? That is just you. So this is a sub-base but it is already a topology. It is fine. If it is not a topology you better make it take the topology generated by that. But here is a case wherein it is already a topology. Okay. So subspace topology is a special case of this. Namely when there is only one member F i is only one and that too we say inclusion map. Take another space now another example which we have done namely the product. Once again you have this product set X i. Suppose each X i is a topology has a topology. Okay. X i tau i it is not X tau i. Okay. Then underlying each X i has a cartage as a topology and they have taken cartage in product of this set. And I have these projection maps P i from X to X i coordinate projections. By definition 2.78 whatever namely product topology it follows that the product topology exists nothing but induced topology. It looks like the definition of product topology has been adopted here to define the induced topology that is all. It is the same definition. Okay. Only this is this is a special case that is all. Right. P inverse P i inverse of U i collection that collection is taken as the subspace. So that is what it is. Okay. So both of them fit into one single generalization here. And this generalization can give you lots of other examples unimagined examples. Not very familiar also. Sometimes wonderful examples. Okay. If you develop some theory for for just product spaces, imitating that it may be available for any induced topology like this. Right. And then it will be available to those all strange kind of things also. So that is the idea of putting this one together right now. This is the deep sea you cannot get into that one right now. So you have just to be aware of it that's all. So here is a remark especially the word weak topology in functional analysis induced topologies play very, very important role. There you have a normal linear space x, one single normal space and take the induced topology on x from the family of all continuous linear maps f from k to f from x to k. Or k is the field. Either you may be real numbers or it may be complex numbers. Why should I put continuous first of all? Why not just linear? When these spaces are large namely infinite dimensional okay linear maps may not be continuous. I have already given you such an example in the previous day in the in the in the yesterday's lecture the previous module okay in the coordinate of the function was n times t. Okay that is a linear map but it is not continuous. Okay so there are lots of linear maps which are not continuous. Linear maps are automatically continuous when you have finite dimensional vector spaces, nonlinear spaces. So you take all continuous linear maps let's say smaller family linear maps into k they are called linear functionals. So with this family now you put a topology on x. Okay so that topology is called weak topology. It's easy to see that with respect to this topology a sequence xn in x is convergent if and only if f of xn is convergent for every continuous linear map. So this was the motivating idea of putting this topology. It is convenient to have this one so that you know we can recognize the space by sequences. Also it turns out that x star which is called the conjugate space of x this is named by some author some people call dual space and so on. Dual when you take all linear maps you take that's why they want to make it different. This is called conjugate space continuously linear maps. Okay so conjugate space x is a normally in space we can give it a norm also. Okay and hence has a topology on it so on. So this norm is what is called the linear norm supremum norm whatever. Okay but often one considers the double conjugate space of x space this x star which is nothing but continuous linear maps you know k linear maps on x star. So all continuously in case x star to k. Take that then use this family to give a weak topology on x star. Okay so in function analysis they have just a different name to distinguish this one but this is also weak topology only this is called weak start topology weak start topology is on x star. The weak topology is on x only that difference is there but both of them are the induced topologies with respect to a certain family of functions so you can just call them as weak topology. There are many other situations also in which weak topology is used in analysis. We can't go into them much deeper. Okay now let us come to coinduced topology. In some in some sense it is a dual notion of induced topology. Okay this time the fact one single y is fixed the co-domain the families of functions from xi to y are taken and each xi is given a topology. Okay so in some sense arrows are reversed that's all that's why it's a dual dual situation. Now again we want to give some topology on y similar to the earlier considerations a meaningful topology meaningful with respect to these functions as well as the topology here. The first condition is that all the files must be continuous once again you can just give induced topology on y the least one very smallest one then automatically all functions should be continuous. Once again this approach is useless so we reject this one outright for a similar reason as we have rejected discrete space in the case of induced topology right so what we want to do automatically brings her to the following definition now let y be a set and f y from xi to y to y be families of functions the largest topology on y such that all f y's are continuous is called the co-induced topology on y from the family f y or you may say with respect to f y okay the following theorem the proof of which is completely trivial or similar to what we have considered in the in the case of quotient spaces or something like that gives the existence and uniqueness of such a topology on y what is the theorem y is a set f y's are functions as as before put tau equal to all u contained inside y such that f i inverse of u is inside tau y for every i okay so this is just very stringent condition we may say but don't put anything else in discrete space is taking is too stringent so don't don't make it too small take only those u's a set f y inverse of u is inside tau y for every i automatically this will be topology automatically whenever there is some top prime here which is such that all the f y's are continuous okay f y inverse of that u will be inside tau that means tau prime is contained inside tau so this is largest okay so that's why this proof is completely trivial is what i told once again let us have a few remarks here one important special case is what we have studied quite thoroughly but we will keep studying it again and again what is it it is one function q from x to y and that is subjective function just start with that as soon as you have a topology here take the co-induced topology on y what is it by definition it is the largest topology on y such that q is continuous what is the construction exactly the same as in the case of quotient space there is no indexing here just one q is there f i f 1 equal to q q inverse of u is open in x will mean that u is open in y over and that is the definition of the quotient quotient topology there where q from x to y is a subjective function and x is a topology so i'm just recalling that one right so so this is a direct generalization you may say of the quotient space construction or if you have done this one before or just now you have done it the quotient space is a special case very very special case only one function and also i'm assuming that it's okay another important case is that we have done that we have studied you know before so this is a little more complicated situation now actually so you have to pay more attention to this one than to the quotient space construction where there is only one function take the case wherein y itself is now union of x i's and y has the coherent topology with respect to x i's which are subspaces remember the what is definition of coherent topology y has a topology in that topology x i's are subspaces and then it satisfies some condition so that is the meaning of coherent topology remember that okay i won't say that this coherent topology is a special case of this case namely now start with a ignore the topology that you have taken right in the beginning for y now give the coinduced topology on y using the topologies of x i's okay and what is the map maps are from x i to y inclusion maps okay x i to y inclusion maps what you get is nothing but the original topology on y so by the very definition of coherent topology it follows that this is the coinduced topology why what is then open subset of y in the definition you see a is opening y a intersection x i is open for every i that is precisely the definition of this one here wherein each f i is inclusion map you contain inside y i will put here if and only if you intersection x i is inside tau i over so it will give you the exactly whatever tau tau you started with so coherent topology is a special case of what induced topology coinduced topology okay however not all coinduced topologies are coherent with the original family of biological spaces x i to y i want to make it clear suppose x i to y's are subspaces okay and you have taken the inclusion maps and now you give the coinduced topology that topology will be coherent maybe but you started already with the topology there right some topology you started and to take the subspaces and then you took to took this coinduced topology that topology may not be quote the original topology or in that sense it may not be the coherent topology so not all coinduced topologies from the original subspaces maybe coherent topology that is the meaning of this a special case of importance which we have discussed earlier when we have a countable family x i's of topological spaces one contained in the other and each x i is closed subspace of x i plus 1 in that case the coinduced topology gives you the same topology which you started with so that means it's coherent topology the coinduced topology in this case we have seen earlier is coherent with respect to x i there are special cases where this can happen like open subsets that's another case or locally finite close sub families etc okay so there are cases wherein this can occur but cases where this doesn't occur off okay another special case of importance is that x i is the family of compact subspaces of a locally compact host or space so this I am mentioning that because it's so important so right now I just mentioned that we are going to study this one later on when we study compact spaces and locally compact host or spaces and so on okay right now you don't know these terminologies let us say so don't worry about this similarly a lot more about quotient maps can be studied only after we get familiar with other topological notions okay so we have just begun these induced and coinduced topologies so we have explicitly five different cases of this okay two four two five or four what how many subspace topology product quotient and this one is union right so and coherent topology and so on union under union coherent topology is under union itself anyway yeah so I will conclude it with a with a easy example here okay but this is only for understanding the depth of our knowledge of coinduced topology all right so take a break that's all this example is not of any use let x to be any topological space okay consider the family eta x x belonging to x what are eta x is our inclusion map of the singleton x inside x inclusion map singleton x to inside x now these singleton x are topological spaces right what is the coinduced topology on the family from this family on the set x what is the coinduced topology that's what I am asking okay by definition something is open in x okay if filled only if intersection with singleton x is open in singleton x for every x if the this this point is not in not in the set u then the intersection is empty so it's open if it is inside that the intersection is singleton x singleton x is an open subset of singleton x so what do I what I have concluded I took an arbitrary subset of x I have shown that it is open right so this will always give you the discrete topology on x right this is a discrete topology on x okay for example more more you know certain example you could have taken x to be r then all singletons will be actually closed subspaces yet the coinduced topology is discrete not the original r at all okay so just closing as it's not enough see in this theorem we had everything is closed in this one and Xi is closed inside Xi plus 1 increasing sequences etc we had right or we had what is called as locally finite family of closets also okay neither of the condition is satisfied here this family is not locally finite okay in general in the inside r for example it is not locally finite so when it is locally finite it just means that the space is discrete space we can prove that okay sorry this is this is locally this is not locally finite but it is point finite okay so this is also a nice example in that way but let us leave it I mean this is just for understanding what happens to the coinduced topology in conclusion starting with a topological space x tau and a cover just like when you had a x to y surjective map in the case of quotient space you take all all the points otherwise some points are left out there will not be any structure on that part so x is union of Xi's of some special importance these Xi's are you have chosen nicely you know represented like they cover the whole thing they are like a representative of subsets it is often the case that we consider the coinduced topology tau hat that is notation 1x which is usually finer than the given topology tau okay so it is a finer topology than tau so this is a an important phenomenon at least to all you know nice mathematical ideas have been developed from this one this can be used by politicians also perhaps in construction of of states and you know countries and so on so here is an exercise now which you can immediately answer because I have explained the things behind it okay let x equal to union of Xi where each Xi is given the topology tau i let tau hat denote the topology coinduced from the collection put tau i equal to tau hat restricted to Xi how the two topologies on Xi compare with each other next thing is show that this tau hat is coherent with respect to Xi tau i small tau i you started with arbitrary tau i okay each Xi it may not be coherent with respect but if you put tau hat equal to tau i equal to tau hat ratio to Xi then it is coherent okay and this this remark this exercise all I already explained it to you x equal to r and I am taking a equal to this family r it is a point finite but what happens you can understand okay so this Xi is already explained you can write down the details so that is all next time we will meet we will meet with chapter 3 okay starting in earnest to study topological properties we have reasonably enough number of examples now other than the standard you know normal new spaces metric spaces and so on okay so we can start studying some topological properties now next time thank you