 Welcome to module 10, having introduced a notion of a homeomorphism and having studied some subsets of R n, let us now do various types of equivalencies coming out of the matrix, we will do it in general now. First of all, the broad aim of topology as a discipline is to classify all topological spaces into homeomorphism types. What does that mean? It means that you must describe a set of topological spaces such that any two members of the set belong to distinct homeomorphism type, that is they are not homeomorphic to each other and second thing is if anybody gives you any topological space then there must be a member in your list which is homeomorphic to that given space. That is the meaning of classification, classification of topological spaces up to homeomorphism. However, this such a task seems to be seemingly impossible. How can you list all possible homeomorphism types? How can you is not the answer? Lot of very difficult things have been achieved by the people. But here is something mathematically established that this is an impossible task. I cannot elaborate that right now because it involves deep logic, deep group theory and sufficiently deep not so deep but beyond you of algebraic topology also. So, beyond this I cannot say anything more than that. It is impossible to classify all topological spaces. So, that sounds rather a negative result and you may momentarily get disappointed oh then why should we study topological spaces and so on. That is not the case. There is no worry for that because there are enough problems other than the central problem or the broad aim whatever you wanted. Many related problems are there in topology. The solutions of which will be tractable. Tractable means what you can just looks like you can try to solve it sometimes solve it also sometimes partial answers and so on. But the final answers to be useful. So, therefore we should not get disappointed that is the whole idea. Let us begin with the definition now. A formal definition which looks like a not a mathematical definition. A property p what do you mean by property p that itself is a difficult thing to explain. So, take it as a you know just with a pinch of salt just take it as it is. Property p of topological spaces is called a topological property or a topological invariant. If every homeomorphism preserves it a little more elaborately suppose you have a function f from x1 tau 1 to x2 tau 2 which is a homeomorphism. Then x1 tau 1 has property p should imply x2 tau 2 has it and conversely x1 tau 1 satisfies p if and only if x2 tau 2 satisfies it. What are x1 and x2? They are homeomorphism to each other. So, that is the meaning of that p is a topological property. So, such a thing is called a topological invariant also. Let me give you some examples then only you may be understanding. Topological invariants are very useful in distinguishing topological types. What do you mean by type? You know you have seen that homeomorphism defines an equivalence relation how these are equivalence classes. Given two topological spaces you find a property p which is satisfied by one of them but not the other. Then obviously they will be in different classes they cannot be homeomorphic. So, this is the way you can distinguish the topological types somewhat easily. Of course, you may find that this property satisfied by both of them that does not prove that the two spaces are homeomorphic because there may be some other property which is not satisfied. You may go on listing all your known properties satisfied by both the sides still it will not prove that they are homeomorphic. To prove that something is homeomorphic somehow you have to produce that is the you know that is the only way. Like we did first we showed all balls are homeomorphic to each other. Then we showed that one single ball namely of radius one is homeomorphic to the whole of r, whole of r n. Now, you can combine the two so that this is the kind of proof we have to produce in proving that given two spaces are homeomorphic. Just topological types will not help in that case. Nevertheless, so they have half the way they come namely distinguish topological type they are helpful. So, before going further I will give you I will take the support to define two things but I will meet them much later. Take a function from x1 tau 1 to x2 tau 2. We say this in open mapping I told you while defining continuous mappings that is such a thing also now I am defining that. It is called an open mapping if it takes open sets into open sets. That is all. F need not be continuous. If take f inverse takes open sets open sets then it is defined as continuous one. Remember that u inside tau 1 should implies f u inside tau 2. Similarly, I can define a closed mapping also namely say k is a subset here x minus x1 minus k is inside tau 1 should imply x2 minus fk inside tau 2 which just means that f of k is closed inside tau 1 should imply fk is closed inside tau 2. So, remember closed sets are defined by taking the complements are inside open sets. So, a homeomorphism is a continuous bijection which is open as well as closed. So, this is one way why open mappings and closed mappings are important. That is why I have introduced it here. You look at this one take a continuous function what is a open subsets in a domain code domain inverse of that will be open. Now, suppose f is a bijection then what does that mean? That means f inverse function is an open mapping. Similarly, if suppose f is both open and continuous then it will imply that f inverse is continuous. Therefore, it will be a homeomorphism. Similarly, if f is open as well as a closed mapping then also it will happen because you may you can see that the continuity can be defined in terms of closed sets also just by De Morgan law. So, I have made you that lots of these things you can change from open sets to closed set by just by De Morgan law where function is continuous if every closed set in the domain in the code domain the inverse image is closed. So, if f is a bijection then f inverse makes sense as a function otherwise f inverse of a set always makes sense as a function f inverse makes sense first of all and it will be continuous because f is a closed map. Therefore, you take a continuous function if it is open it is already a homeomorph and sorry continuous bijection. Similarly, a continuous bijection which is also a closed mapping is a homeomorphism. So, why sometimes it is easy to handle closed sets sometimes it is easy to handle open set that depends upon the situations that is why we should have all these concepts clear to us. Now we will come to metric spaces wherein there are other equivalences other notions of equivalences just like we have in your school geometry you have congruent some congruent triangles as well as similar triangles and so on. They are both equivalences if one triangle is congruent to another triangle the another one is congruent to third one then the first one is congruent to third one and so on we know that congruence as well as similarity are equivalence relations. Similar to that we will have different notions of equivalences here but they are all metric related. So, let us start take two metric spaces X i d r. We say they are topologically equivalent if the underlying topological spaces X i tau d i's they have the same homeomorphism type that means there is a homeomorphism between them that is one thing namely topological type topological equivalence. The second one we say we say X 1 d 1 is similar to X 2 d 2 if you have a bijection from X 1 to X 2 and two positive real numbers C 1 and C 2 such that you know this d 2 distance between f x and f y is trapped between d 1 distance is between x and y by these constants C 1 times d 1 is less than or equal to d 2 of f x f y less than or equal to C 2 times d 1 again. C 1 and C 2 must be positive real numbers. If C 1 and C 2 are one what is the meaning of these they are all equal that is a very strong thing of course that also we will take that is an isometry. So, we will come to that also. First of all why this similarity is an equivalence relation you see this looks like one way I have defined f from X 1 to X 2 but it is a bijection. So, there is a map f inverse from X 2 to X 1 then I can write f inverse here and X y I take now instead X 2 then what should I take C 1 C 1 now C into C 1 I have to take d 2 d 2 here d 2 of X y and here also d 2 of X y what should I take you just look at this one what should I take to bring C 1 here it is 1 by C 1. So, then d 1 now if I can write this as f of A f of B f inverse of f inverse B then this will become AB. So, work it out I have already indicated how to do that ok. So, you have to use 1 by C 1 and 1 by C 2 in correct places and write down some similar thing for f inverse that will show you that the similarity here is what is symmetric ok. Then you have to show if f from X 2 y X 1 to X 2 then from another g from X 2 to X 3 and there will be C 1 prime and C 2 prime and so on. So, you have to get a g composite f from X 1 to X 3 as a similarity. So, that will show you transitivity of course I can take identity map here and C 1 equal to C 2 equal to 1 that will give you that identity map is a similarity. So, therefore, this similarity having a similar map having a similarity like this a bijection is an equivalence relation you have to do a little bit of verification. So, that I am leaving it as an exercise to you, but I have already indicated it and given you hint enough ok. So, this equivalence relation is a stronger than the first one why because as soon as we have this one you will see that f is continuous ok. So, this part will give you f is continuous. So, that part will give you f is f inverse is continuous. Therefore, it is homeomorphism. similarity already implies topological equivalence. So, it is similar to it is you know a stronger equivalence here. So, that implies weaker equivalence that is all ok, but they are different is what I have to ensure you that we have yet seen ok. Let us go ahead. So, I have already told you if you put C 1 equal to C 2 what you get you will get that distance between f x and f y that d 2 distance is equal to distance between x and y the points that which you started with ok. So, such a thing is called isometry. So, it is a bijection f is a bijection that is already there the distance between x and y is preserved under f d 2 of f x f y it is equal to that ok. If there is such an isometry then you will call x 1 d 1 isometric to x 2 d 2 this is very easy to see that this is a equivalence relation ok whereas, in the similarity you have to work a little hard ok. So, I want to caution you that some authors may have different names for these things ok. So, be careful to read their definition before answering their questions or whatever. Clearly isometry implies similar and similar implies topological equivalence. So, there are three concepts which you have now starting with metric spaces one is stronger than the other this implies this this implies that, but are they really stronger maybe this is this will also imply that then they will be equivalent now. So, let us see that you may notice that a self isometry of R n is nothing but a rigid motion, but now by now you must have you must have at least read and tried the exercise on rigid motion then only you will understand this remark ok. A rigid motion was defined as what any function which preserves a distance like this there was no condition of bijection. The only thing is the domain and co-domain were both assumed to be R n that is no need you can assume this one to be anything any topological space, but the same topological space on this side any sorry any metric space and same metric space on this side also ok. Rigid motions are usually within a metric space. Take x1 d1 to x1 d1 and map such that distance between xy equal distance between fx and y for our xy. Automatically you can check that it is injective, but on to-ness is an exercise or so far you do not know that it is on to and so on ok. So, that was the exercise. So, this is then stronger than a rigid motion when x1 and x2 are the same x1 d1 x2 d2 must be also equal to x d1. So, it is not different then the word rigid motion and is used. So, rigid motion is a weaker motion than this one then isometric. Isometry is definitely only thing is this is valid for between any two metric spaces ok. So, here is a theorem take two metric spaces which are similar to each other then they are homeomorphic to each other. I have already told you that I will elaborate it on that one namely start with a bijection and c1 c2 positive such that this is true for every xy right. So, that is the definition of similarity. I have to show from this one that f is continuous and f inverse is also continuous. We claim that f itself is a homeomorphism. Then it will follow that x1 and x2 are homeomorphic that is all ok. So, we have to prove both f and f inverse are continuous because bijection is already given. Notice that condition 11 can be broken up into two parts namely this latter part d2 fx fy is less than or equal to d2 of sorry c2 times d2 d1 of xy for every xy inside x1 ok. Second part you take inverse here d1 of f1 inverse a right f inverse of a into f inverse of b xy equal to this one ok bring c1 on this side 1 by c1 of the d2 of now f of that will be a a comma b for every a b inside x2 ok. I am cleverly writing this using the using the definition using the fact that f is a bijection. So, f inverse makes sense. So, suppose you have these two conditions you can give you back these conditions. So, I am writing this condition one condition is equivalent to these two conditions here ok it is broken up into two parts right. Once you have that by symmetry if I prove this implies f is continuous automatically this will imply f inverse is continuous ok. So, this 12 and 13 these these two conditions are there suppose this 12 implies f is continuous then the other one is off. What is here given so how do you prove continuity of f given epsilon there exists a delta blah blah blah right that is what you have to do. So, whenever distance between this is less than delta this must be less than epsilon if this is less than delta this entire thing will be less than c2 times delta. So, I have to only say that c2 times delta is less than epsilon then this will be less than epsilon. So, that is what I do here given epsilon positive choose delta such that delta times c2 is epsilon or delta I put epsilon by c2 then d1 of xy less than delta will imply d2 of fx yf fx fy here if this is less than delta this c2 delta which is epsilon this is less than epsilon ok over alright. So, one equivalence relation other than homeomorphism you have studied in fact two of them, but one we have shown that similarity implies homeomorphism. The other one is easy right because a special case isometry if you take c1 equal to c2 equal to 1 here you get to similarity. So, isometries imply similarity ok. So, next time we will see that homeomorphism does not imply similarity and similarity does not imply symmetry ok So, we will do it next time thank you