 Welcome to a modulate of point set topology course. So in mathematics, the study of objects always goes hand in hand with the study of appropriate functions between them. Like when you are studying vector spaces, you will take linear maps, in your studying groups, you will take group homomorphisms, right, like this when you are studying metric spaces, we started with continuous functions epsilon, delta definitions. So we want to extend that definition to encompass all topological spaces, okay. So the new definition that we are going to make always should encompass the older definitions, okay. That must be the motivation of keeping this and it should give you more. So that is the whole idea. So let us make this definition first, namely of continuous functions between two topological spaces x1 tau 1, x2 tau 2 or two topological spaces, a function from x1 to x2 that is a set theory function, it will be said to be continuous, okay. So it will be called continuous function provided inverse image of every open set in tau 2, namely in x2, should be open in x1, v is in tau 2, f inverse of v must be in tau 1, okay. So for the first time, you may see that why things are happening the other way around, but soon you will realize that this is most natural way to define, it is not open sets go to open sets, inverse image of an open set is open, that is the more natural thing. There is a concept which is open set goes to open set, that becomes a subsidiary concept which is not so important as continuous functions, okay. So there is another one also, so we will come to that one later. This is the correct thing in terms of epsilon delta definitions of our metric spaces. So let us see how that is true, okay. So that will be justification for making such a definition in the case of a logical spaces. This tau 1 and tau 2 are just a logical spaces, they might not have come from any metric. But suppose they come from metric, then you have two different definitions, one is for this one whatever you have given just now as to power of v, but there is already something called continuity coming from metric definition. So what we want to say is these two things are coinciding, okay, so that is the next theorem here. Let X1, D1 and X2, D2 be two metric spaces. Take a saturated function from X1 to X2 as usual, that will be a continuous function in the topology tau D1 to tau D2 here, now these are topological spaces. If and only if as on metric spaces XD1 to X2 D2, it is continuous, namely where it is epsilon delta continuous, for every point given any point X belonging to X1 and in epsilon positive there must exist a delta such that D1 Xy less than delta implies D2 fx fy less than epsilon, so this is I am just recalling here, I am not making another definition here of course. So this is the epsilon delta definition for a function between metric spaces. It is equivalent to the continuity of the same function on the corresponding topologies induced by the metric, okay. So once we prove this one, there will be a full justification for this definition, alright. So let us just prove this one which is not at all difficult. Assume that f from X tau D1 to tau D2 is continuous according to the new definition, okay. Take, now I want to prove it for epsilon delta. So take epsilon, epsilon point is given, X is given, epsilon is given, right. Look at B epsilon of fx, that is an open subset B inside X2, alright. This B epsilon of fx is a member of tau D2. So f inverse of that must be inside tau D1, so in tau D1 it is open. But now look at here I have taken fx here, fx is a point here. So in the inverse image X will be there, therefore X is in f inverse of V and f inverse of V is open. So it follows that by the definition of the dystopology that there is a delta positive such that the delta ball around X, okay this is an open ball is contained in f inverse of V, right. Because f inverse of V is the union of open balls inside this metric space tau D1 and metric space D1, alright. So it must be such that B delta of X must be in f inverse of V. Now D1 of XY is less than delta implies Y is in B delta of X, right. D1 of delta XY, D1 of XY less than delta means this Y is inside here. So Y is inside, f inverse of V means f of Y is inside V, f of Y inside V means what? Look at this one, this Y is in V delta of X and hence fy is inside V, so fy is inside V means D2 fx fy, the distance span must be less than epsilon, one way is done. Conversely start with any V inside tau D2, we have to show that f inverse of V is open namely it is inside tau D1 is what I have to show. So to show that this is inside tau D1, take any point X inside X which is inside f inverse of V, that means what, fx is inside V. But V is open, fx inside V means you must have epsilon positive such that B epsilon of X is contained inside V, therefore there is a delta positive such that you know that is an epsilon definition as in the statement that means what difference between X and Y is less than delta would imply fy is inside this open bar, okay. So choose such a delta, then f of B delta of X is inside V, therefore B delta of X is inside f inverse of V, so this is true for every X f inverse of V is open inside X1 namely tau D1 it is in the measurement of tau D1, okay go through this proof carefully. So here I have used the fundamental property of tau D1, tau D2 that around every point inside an open set there is a ball around that which is contained inside that, so this is the property I have used here, okay. Now one remark here is unlike for the metric space is wherein we have the notion of uniform continuity, we do not have such uniform continuity concept in the topological spaces, okay, there is no such notion except you have to work hard namely we will do that later on remedy for some smaller class of topologies which are not necessarily metric topologies, if it is metric topology of course we have, okay. We need to put extract structure, uniform structure on that domain and co-domain what are called as uniform structure, so they are not ordinary topological spaces but satisfying more conditions, okay. So that will not be done in this course, okay it is not a main thing it is a side topic, so we will not have time for that, analog of theorem 1.19 is true in the general case of topological spaces and it is easier this 1.19 is nothing but theorem on composites, okay. So this remark was a negative thing it is all, this remark was somewhat in the negative root on but rest of them will be now very happy things, everything positive. So composite of continuous functions is continuous, theorem 1.19 for metric spaces, same thing is here true and proof is much easier now, see it is much easier what you have to do, there is a function here, the function here the composite is there, right, take an open set here inverse image open here is what you have to do, inverse image here under G first comes here but that is an open set because G is continuous, now you take the inverse of that that will be the full inverse image of under GF of inverse, okay GF inverse of U is G inverse U and then F inverse U, right this is the set theoretic property this is pure reset theoretic property, U is open, this is open, this is open, F inverse of that is open. So it is easier to show that composite of 2 continuous functions is continuous in the case of topological spaces, alright therefore this also proves now whatever we prove for metric spaces, you see we need not have proved that we have used that, okay a continuous function X1 tau 1 to X2 tau 2 now arbitrary topological spaces, okay set to be a homeomorphism this is the word I am going to use now, okay where it is what is a homeomorphism, F is a bijection and its inverse is also continuous, F is a continuous function it is a bijection therefore it has an inverse that inverse is also continuous, okay suppose you have a homeomorphism from one topological space to another one then the two topological spaces are called homeomorphic to each other this is homeomorphic to this, okay from the previous theorem it follows easily that being homeomorphic is an equivalence relation on the collection of all topological spaces, one thing is clear by the very definition homeomorphism, inverse is also homeomorphism there is no need to work because it is a bijection, inverse is a bijection, inverse is dead, inverse is continuous, F is continuous, so F inverse of inverse is F itself therefore if there is a homeomorphism like this F inverse will be homeomorphism from the other way around so symmetry comes any topological space is homeomorphic to itself because identity map is always continuous and bijection of both ways from the same topological space to same topological space, okay identity map is always continuous no problem therefore reflexive, transitive is what you have to say transitive it is precisely this theorem F this one is homeomorphism this one this one is homeomorphism to that one X1 to X2 to X3 so X1 to X3 okay so homeomorphism is a function being homeomorphic their topological space is so on topological space is this is an equivalence relation okay so this equivalence relation is of profound of interest to us any two closed intervals so here are examples any two closed intervals consisting more than one point is a closed interval could be singletons so you should avoid that singleton is a singleton if there are more than one point then that closed interval no matter what it is all of them are homeomorphic to each other this is the statement similarly any two non-empty open intervals may take one-one empty rather than non-empty they will not be homeomorphic okay empty set is never bijectively to any other set okay so any two non-empty open intervals in R are homeomorphic so there are many homeomorphisms actually but you can find something which is very nice namely of the type fx equal to ax plus b is a linear polynomial polynomial of degree 1 so when is this a homeomorphism it must have an inverse right it is very clearly that a must be non-zero b could be anything okay then you can write down its inverse these are linear maps okay for any non-empty open interval is homeomorphic the whole line itself the interval is bounded the whole real line is not bounded but still they are homeomorphic to this is what one has to see and there in several ways you can see standard maps are the following look at fx equal to x divided by 1 minus mod x so where I am going to define from the open interval minus 1 to plus 1 to the whole of R suppose I have proved this one a homeomorphic then I know any open interval finite like this will be a homeomorph to any open interval ab by this method so all of them are homeomorphic to the whole of R that is what I get so how do you get this homeomorphism very easy look at its inverse it is nothing but x divided by 1 plus mod x okay you can compute it the standard method put y equal to this and solve for x in terms of y so you may have to because there is a modulus you may have to make two different cases x non-negative and x negative okay so if x is positive what is this is 1 minus x then you can graph y equal to x divided by 1 minus x you can rewrite it in terms of x equal to something in terms of purely in terms of y and so on so that is the way to check this one is a homeomorphism easy way directly write down the formula for the inverse okay here another one from trigonometry tan pi by 2 x x is ranging from minus 1 to plus 1 so the domain when you put minus pi by 2 it goes to minus infinity plus pi by 2 to go to plus infinity 0 goes to 0 so it is a strictly monotonically increasing function from minus 1 to minus 1 1 2 to 0 1 because you can look at its derivative blah blah blah that is why it is a trigonometry some cal plus unit okay to show that it is and then you can show that this strictly monotonically graph will be like that going like that right so both minus 1 plus 1 go to this goes to minus infinity goes plus infinity those points are not there but the entire all the open all the whole values in r will be taken it is a subjective map because 1 goes to infinity and minus some goes to minus infinity everything in between must be there by intimate value theorem there are so many different things you can use so why this is a homeomorphism I am telling you that is why you can write tan inverse there is a justification for writing tan inverse okay given any topological space x you can look at all self-homomorphism from r to r from r2 to r2 and so on take any topological space take a map from x to x which is continuous and bijection inverse also continuous look at all that to compose them again that will be again a homeomorphism to take the inverse that is again a homeomorphism identity map is always a homeomorphism these three things together what do they make they make a group that is the definition of a group is in fact the group of homeomorphisms and automorphisms and such things they are the the harbingers they are the originators of group theory okay the self-homomorphism forms a group unfortunately this group is too huge unlike in group theory in the beginning you get to know you know small small groups or nice groups like integers and so on okay so in general study of this group namely HS that brings out the geometry inside x okay in fact people have gone to the errant of defining geometry as the study of the groups of homeomorphisms groups of automorphisms groups of isometries and so on or subgroups of these groups what is happening in this group that is the geometry okay this group is quite huge let me elaborate what is the meaning of this quite huge a little bit okay such study cannot be completed in any semester course okay so let us look at some examples here take all linear maps t going to lambda t plus mu or at plus b like ax plus b i have written that lambda and mu are real numbers lambda must be not 0 is all that you need to show that this is invertible now i am writing the inverse also inverse will look like s going to s by lambda minus mu by lambda you can check that this is the inverse okay so they are all there they are all homeomorphisms of r with itself given two pairs of real numbers a1 b1 a2 b2 okay a1 less than b1 a2 less than b2 b2 is whichever way you want so you assume that there is always a linear map alpha which sends a1 to a2 and b1 to b2 you just write down a linear map ax plus b alpha x plus beta and solve for alpha and beta by putting this condition so that is that is still standard stuff solving two simultaneous linear equations so that will give you the formula for this function itself okay namely this alpha t is now the function which takes a1 to b1 and a2 to b2 check that there may be some errors here you should check that okay so if there is a b1 is here become b2 that is not all that question you have to check that and correct it if at all if it is correct it is fine so what is claimed is that you can solve for this alpha t something like a a t plus b i am finding this a and b here what is the condition a i should go to b i a1 should go to b1 a2 should go to this very straightforward all right once we have done that this is already a geometry see I can find a map which is a bijection one one mapping one one correspondence from any interval to any other interval right there is a meaning of this one now okay but there is more than that okay more more geometry is coming out take any set of real numbers finite set of real number put them in an order take another with the same number put them in the order the order is important here when you are taking only two of them even the order was not important you could have mingled here a1 going to a b2 and b1 going to a2 that is also possible here but here it is it is important that you should have the same order then you can find a homeomorphism f which takes a1 to b1 a2 to b2 a n to bn okay how do you get that I will explain it to you I will explain it to you I do not want to write down the full formula if you want you can write down actually I have written down that also but first I will explain it to you these are a1 a2 a3 a4 this is b1 b2 b3 b4 here okay I want a1 to go to a2 means the graph of that the point will be here right this is a graph a2 just go to b2 so the point will be here a3 we go to b3 point would be here like that so these points I would then I would join them by straight lines because I know there is a linear map which takes ai to bi ai plus 1 to bi plus 1 concentrate on each each interval here first two at a time right take two at a time in the order namely so this interval that interval that interval first make take get a map which takes this one to this point and this one to this point so this is this map and this is the line segment here there is no condition so extend the same line segment don't disturb it at all here there is no condition beyond that extend that line the same way all the way from the last line but in between join them by line segments determined by those points this is a graph you can write down the formula now no problem so in each ai to bi they are very different formula okay so once you know this we can write down I have written it down here you can check it there may be some errors here ai may be missing ai a2 may be seen something i may be become i plus 1 all that you have to check okay so this is not a big game you have no of course but what you should understand is how to get this one all right so I have defined the function in three different ways but this this middle one gives you all the in in between intervals the last two gives you what you have to do namely don't worry at this part you have to extend it as if we have defined here similarly whatever function comes here you extend it here in between use the from i a2 i plus 1 use this formula so that's what I have done here okay f i of t various f i of t's are there right we send ai to bi so I have taken this f i is function f of ai to bi and f i of ai plus 1 to bi plus 1 for each fixed i there is an fi if you change i of course they will change right that's correspond to these having different slopes here okay so indeed here is a theorem from real analysis which characterizes all elements of h hr the homeomorphisms what are all the homeomorphism this is a characterization it may not be much helpful but it is quite helpful a function f from a to r is a homeomorphism if and only if it's continuous and strictly monotonic strictly monotonic because you want you want what huh you want it to be one one mapping right like that okay it may not be on to but you put on to also it will be homeomorphism on to r otherwise it will be homeomorphism on to the image okay so I am using this word here only slightly more general here that's all see I have not put on to next year I should put on to next year also okay yeah so let us stop here today next time we'll do more on not just on r but now r power n that will be the next topic thank you