 Welcome to module 23 of point set topology part 1 course. So today we will carry on with sub-basis. We have motivated the definition of sub-base and so on last time okay. So let S be any family of subsets of S, take P to be the collection of all finite intersections of members of S. First of all S will be there inside P, all members of S there will be there. Two members you take their intersection will be there, three intersections will be there, like finitely many members, their intersection will be there inside P. So that is the B. By definition B is that collection, nothing more, nothing less. Then B is a base for sub-topology on S. In fact what is that topology? It will be precisely tau S which will be tau B okay. That part we have seen that it has to be tau B okay. So I do not have to state that one. So what we have to verify? We have to verify the two conditions B1 and B2. I do not have to worry about describing tau B here. The tau B is already described okay. If you take empty intersection, it follows that X will be inside B. S, S itself may not contain the element X you see. But when you take finite intersection, empty intersection is also allowed here okay. So that will give you X inside B. So B1 is automatically satisfied in a strong way. See B1 was that union of members of B is equal to X. But here I am putting X itself is in fact, this is stronger way okay. It is stronger than our B1. Now given two elements inside B, intersection will be inside B because this is some say intersection of say BI1, BI2, BIAK. This will be B21, B21 and so on. Intersection will be you take all of them. So it will be this is say 5, this is 6 and that will be 11 like that. So intersection will be also inside B okay. So I can take B3 itself as given intersection B2 to get the condition B2 right. So what is condition B2? Given any point in the intersection, there must be a third one which is which contains that point and contained in the intersection. I can take the intersection B2 itself because that is the member here. So this is both B1 and B2 are satisfied in a strong way. In any case, this is a base for some topology and that topology is nothing but TOWS. Note that the collection B arising out of S has above satisfies slightly stronger condition than necessary. So it is closed under finite intersections right. So for example, we have seen that the family of all open walls in XD is a base for TOWD. It does not satisfy the strong condition because intersection of 2, so in order to give proper recognition to this result 2.11 lemma whatever we are here okay. So what we do is we will make a formal definition. Let S be any collection of subsets of S at X okay. X is fixed at then you take a collection of subsets of S. We say S is a sub base for that topology TOWS okay. So this definition may look funny but it is what it is just like a nomenclature. We do not put any condition on S. So any subset is a sub base for a particular topology. What is it? TOWS the unique topology okay which is the smallest topology containing S okay. So I repeat thus every collection of subsets of X is a sub base for a unique topology whose members are described very nicely. What are they? First look at the base. What base is given out of S? Namely take finite intersections of members of S okay. So each of them will now each of each of members of TOWS will be in arbitrary union of members of B which are just finite intersections of members of S okay. So it takes no extra condition for the family S to be a base, sub base for a topology. Also as seen in 2.72 part 2 if B is a base for TOW then every member of TOW is a union of members of B. So every base B for a topology is also a sub base because TOWB is the smallest topology containing TOW, containing B okay. So TOW is TOWB means B is a sub base also. So B is a base is something more stringent but TOWB equal to TOWB immediately implies that B is a sub base there is no problem okay. The converse does not hold all base not all bases, sub bases are bases because B2 condition may not be satisfied even B1 may not be satisfied okay. Look at this example. The collection is all unbounded intervals inside R. What are they? A open to infinity or minus infinity to A open. Look at all these this collection is a sub base for a topology. This is not a base for a topology neither it is a topology okay. The sub base for a topology and that what is that topology? That topology is same thing as the usual topology. Why? Because if you take 2 members like this okay A less than B then take A to infinity intersection with minus infinity to B so that will be the open interval AB. So all open intervals AB are finite intersections of members of S and then we know that all open intervals form a base. So therefore S is a what? S is sub base clearly it is not a base. So it follows that members of tau is an arbitrary union of finite intersection of members of S. I repeat this one this is what you have to by heart maybe. Note that a topology can have many bases and sub bases but a sub base and the corresponding base always use the same topology. Okay. So clearly S equal to B if S is already closed under finite intersection then if you take finite intersection further you do not get any more members. They will get members of S only. In general every base is also a sub base but the converse does not hold as seen the previous example. So I am just repeating all these things in this part I have not said anything new. Now I will make a theorem here okay which will show you the utility of basis and sub bases. A function f from x tau to x prime tau prime okay is continuous if and only if f inverse of u belongs to tau for every member u in S prime where S prime is a sub base for tau prime. If it is continuous take a member u in S prime it will be already in tau prime. Therefore by definition of continuity f inverse of u is inside tau. Inverse image of an open set is open that is what you have shown. Conversely we have only partial condition here. Inverse image of open set open holds only for elements of S prime but that is enough because once this is true for S prime it will be true for intersection of two of them. It will true for intersection of finitely many of them that means it is true for members of b now okay members of the base b but then if you take inverse image of the union is also union of the inverse images. So this is purely set theoretic fact you have to use. Inverse image of the intersection is intersection of inverse images. Inverse image of the unions is union of the inverse images. So first use this one then use this one to get all the members of tau prime their inverse images are open. Therefore f is continuous. So what is the role of this theorem? It reduces the study of a continuous function checking the continuous function. You have to do it for all elements of tau prime no. You have to just do it for h value chosen sub base that is enough. Okay for example you can apply this to this theorem here this base here this example okay you have continuous you have some function r to r to check that it is continuous you have to just show that inverse image of unbounded intervals are open both minus infinity to a as well as some a to infinity for all of them if inverse images are open that is enough. You do not have to worry about all open sets okay so we will come back to this phenomena again. So a few more comments about basis and sub bases if tau 1 and tau 2 are two topologies b1 and b2 are respectively their bases okay you have chosen some bases. If b1 is contained in b2 automatically tau 1 is contained inside tau 2 why because members of tau 1 are unions of members of b1 therefore they are member union members of b2 also therefore they are in tau 2 that is very easy right okay what is more useful is you do not need b1 will be contained inside b2 if b1 is contained inside tau 2 then tau 1 is contained inside tau 2 because tau 1 is the smallest topology containing b1 right so this is quite useful exactly same thing is true if you replace b1 b2 etc by s1 s2 which are sub bases same reason I have written down this one here but that is obvious so once again for later use I will introduce these two definitions here tau 1 contained in tau 2 okay any families I have such okay not only for topologies here I have defined for topologies just means that tau 2 is finer than tau 1. So this is the definition of the word finer here in the same time tau 1 is coarser than tau 2 it is just like saying that tau 1 is less than tau 2 or r1 is less than r2 implies r2 is bigger than r that is all so coarser and finer so you have to get used to these terms this can be used for any families of subsets of x of the same x so in particular I have given this definition for two topologies no problem here are the examples discrete topology on a given set is finer than every topology on that set exactly same way in discrete topology is coarser than every other topology on that set okay if x has more than one point then in discrete topology is strictly finer sorry discrete topology is strictly finer than the in discrete topology so stricter inequality also holds there no problem okay so I come back to this example again here the left-rate topology and right-rate topology okay introduce in example 1.18 or we have 2.18 the collection l r the recall namely minus infinity to a a a belong to r is a sub base okay for l r not for usual topology for usual topology you have to combine both of them you take l r itself only l r that will be a topology and that topology we have taken is a sub base for l r and it contains all elements of the topology except empty set and the whole set this is the strangest thing why because look at this one take intersection of minus a to minus infinity to a with minus infinity to b what it will be depends upon whether a is smaller or b is smaller okay if a is smaller intersection will be minus into a so it is remember so this family is closed render finite intersection it is also closed render arbitrary union so what is missing here to be a topology what is missing here empty set is not there the whole r is not there that is all the empty set and the whole set are not there otherwise it is already a topology right so such a base or such a sub base is actually sub base right because if the union is not the whole of x it is not a base so it is this is not even a base it is it is it is not topology but it is a sub base a very string kind of similar remark holds for r r also okay you can compare these two topologies with the usual topology okay what do you conclude I don't want to tell you you can you conclude your thing this is obvious but you have to keep doing this kind of thing that is why I have put it this all okay now let us make a make a definition and rate it to why these things are important okay note that a function x tau to r comma l r this is left raise right it is continuous if filled only if set of all points x belong to x set f x is less than a this is open in x x is any topological space here I have taken r with l r okay so what is the condition for a function to be continuous inverse image of a basic or sub basic open set is open that is enough sub basic open set is all points minus infinity to a inverse image will be just all effects set f x is less than a if a function satisfies this property it is called upper semi continuous function so this terminology is taken from analysis similarly if you replace this one by r r okay then what will be the condition for continuity f x bigger than a must be open okay exactly similar if that happens we will call it as lower semi continuous function so these are important in analysis especially in measure theory so I have included a few exercise exercise there are examples a few few properties of this level semi continuous functions okay so go through them they are not all difficult so x i1 x i1 gives you an example namely this is called characteristic function of a set chi a of x equal to 1 if x is a equal to 0 if x is not a okay so characteristic functions are lower semi continuous if filled down means a is open upper semi continuous function if you notice a is closed these are highly what you may say discontinuous functions but they have some continuity property so that is very important in measure theory you starting point of you know defining measure and all that okay so here are something supreme of any collection of lower semi continuous functions is lower semi continuous infimum of any collection of upper semi continuous function is upper semi continuous if something is continuous then it will be both lower continuous and upper semi continuous and conversely okay these are all elementary exercises to take some time to do them don't leave them because if you ignore them soon these things will ignore you you you give them proper you know time and then they will become friends to you okay so thank you we will meet next time again