 Welcome to module 6 of point set topology course. Today at last we will start the definition of topological spaces and some few examples. Before that let us motivate this definition by studying one important aspect of same thing in metric spaces. So this is, I have put it as a theorem. You can say this just to surmise what is happening in metric spaces. Let X to be a metric space, put tau d, this is a collection of subsets of x, u contained such that u is union of balls, open balls, b, r, i, x, i remember are open balls with centered x i and radius r i. So I have put union of that, I do not know how many I have taken here, that is why I have u i. It could be finite, it could be infinite, it could be uncountable, whatever it is. It could be empty also. When it is empty, what is the union of empty? Empty set is empty set, where it is also allowed. Take all subsets u, it can be written as union of balls. So that is my TAD. Then this satisfies the following properties. First thing is, just now I told you empty set is there because it can take empty union. X is also there after all. Why? Because once any point x belongs to x, it will belong to some open ball and then that open ball will be there. So you can take union of all open balls around each point that will cover the whole of x. Therefore x can be written as union of balls like this. So x is there. If you take a sub-family of tau d, any sub-family, then the union of that sub-family will be also in tau d. The third thing which is very simple-minded but very important, most important one. If two of them are there probably, then the intersection is there. Once this is true, it will follow that if finitely many of them are there, then their intersection is also there. So from 2 to 3, 3 to 4, any finite number we can get. So these are the fundamental properties, I have just collected them. I can collect a number of them, lots of them but these are the things which I am concentrating. Just like we started with modulus and took three important properties. We made it into a norm. Similarly, three important properties were made into metric and so on. So along with the same theme, we have taken these three important properties. So let us first test them. These are going to be axioms finally. So let us call the members of TD as open sets. This is just a tentative definition. Soon we are going to change the definition of open set. So right now let us call them as open sets. It is not the same thing as open balls. Open balls are defined in a different way. Now an open set means union of open balls. Let us call them as open sets. The theorem says that the whole space and empty set are open. Arbitrary union of open sets is open and finite intersection open sets is open. Given a sub-family, one part I have already explained to you, part one. I have already explained to you why this happens. Given a sub-family, when you take the union of over all of them, union of u alphas, each of them is a union. So you can write it like this. Union over union is just a union. It is like a double summation. Wherein in the case of numbers, you will have to bother about whether it is convergent and so on. Here there is no question of convergence. Double summation, whichever way you want to take, you can submit up. It is like union of sets. That verifies to. The third one requires a little bit of analysis. What is this? Namely, you have to use the property of the distance function appropriately. So take a member here, take a member here, u1 and u2. They are themselves union open balls. When you intersect them, what is that? That is the intersection of, it is a union of intersection of one ball here and one ball there. One ball here, one ball there. All pairs of intersections you have to take. Some of them may be empty, some of them non-empty. But it is a union, intersection of the union something, union something is union of the intersection. That is what I am going to use. Therefore, it is enough to prove that intersection of just two of them, okay, two of the balls now is again a member like this. It is a union of balls. See, first I have to prove that intersection of u1 and u2, where u1 and u2 are arbitrary open sets like this, arbitrary unions like this, intersection is over. But that will convert it into just proving the intersection of only two members. Namely, how they are themselves balls now. u1 is BR one and u2 is BS. So I show that this intersection is a member of TD, then the union of all these will be the arbitrary interaction. So that will be out of there, okay. So if u1 is BR of x and u2 is BS of x, that belongs to the intersection would imply both dzx and dzy are r, this is less than r, that is less than s. That is the meaning of this, right. That is in this ball means this one, that is in this ball means this one, right. Now what is that I am going to look at here, look at this one. This is a ball of radius r, that is a ball of radius s around y. Now z is taken in the intersection, you can see that the intersection is not a ball, okay. Here it is a lens, it is a lens like thing, alright. But it will contain a small ball around z for each point, it contains a small ball, so it will be a union. What is that ball? You have to choose this radius appropriately. If the distance from x to z is a and the radius is r, whatever is left out here is precisely r minus a. So I can take as big as r minus a. But I have to worry about the other one also. Here radius is s and distance is b. So I have to take at the most s minus b, therefore I have to take the minimum of the two. So I take the minimum of the two, then I am done. That is a positive number and that ball will be contained inside both of them, okay. That is precisely what I have done here. h d z of x, b is d z y and epsilon z is minimum of r minus a and s minus b, alright. Then if p is inside this ball, you have to use triangle inequality, distance between this and this, okay, is less than this number. It will imply the distance between here to there which is a plus distance here that will be less than r, similar with the other one. So that is what I have done. p belongs to b epsilon z implies d z p is less than epsilon z. But epsilon z is minimum of r minus a and s minus b, so it is less than r minus a. So I can write here if you take exactly minimum less than equal to correctly to be sure, does not matter. Less than equal to is good enough. Implies b p z, okay, actually less than z because here itself I am taking less than, so this epsilon, but this may be equality here, but this is less than, that is correct. p d p z plus d z, I have this one. I am replacing r by this one, taking a on this side to less than r. When you take a on this side, it is less than r. So d p r is, d p x is less than r, so p is inside b r. Similarly, p will be also in the bs of x, therefore it is in the intersection. Therefore, so finally this is what is, intersection of these two balls is union of all these bs epsilon z, this epsilon z will depend upon z. Take them all inside in the intersection, so it will cover the whole thing. So this union is one, so that will be inside k d. So now the whole idea of doing this one is we have these three axioms here, they become axioms here or at a polish. So let us take any set x, take the power set x, take a family tau of subsets of x, that is a subset of p x itself. This should be called a topology on x if it satisfies the three axioms which I have denoted by now, t, a u and f i. x and empty set are inside tau, that corresponds to the first one there. The a u is arbitrary union. If f is a subfamily of tau, take all members in f, then take the union, that will be also in tau. If f is a finite subfamily of tau, take the intersection over this finite family, that will be also in tau. Instead of writing u and u to u n, I have just tried f is a finite subfamily. So these three axioms are there now. A set x together with a topology tau will be called a topological space. Logically, we should write just like we write x, d for a metric space, we should write x, tau for a topological space. However, in practice, we will cut down the notation just like in the case of metric spaces or non-linear spaces and so on. We may merely say that x is a topological space when the tau is understood. If we are discussing two different topologies on the same set x, then definitely we will write it. Sometimes there may be a different topology on a different set and so on that on those things you should write, otherwise we need not write. Writing is not a crime. Only if you keep writing it, it takes more time and more space and so on. Somehow when there are two of them, you have to write which topology. So let us continue with the definitions. Elements of tau are called now open sets in x tau or merely open sets in x. When you say open set in x, we mean the topology is already understood, it is behind. In particular axiom one says x and phi are open. In every topological space. Next, we will make a definition a subset is called if and only if it is complement. Sometimes I write complement for xc, fc. I prefer to use this longer definition, longer notation x minus f. By the way, this you can read it as minus f but do not confuse it with the additivity of our real number or complex number. It is not a minus of that. So should not write the horizontal minus here. So this synthetic minus is the correct thing. If the complement of f is open, then you call f closed set. Compliment of an open set is closed. Compliment of a closed set is open by De Morgan law that is all. Clearly, knowing all closed sets is the same thing as knowing all open sets. If you know all the closed sets, you know the complement of that is all. Often using De Morgan's law, a statement about open sets can be converted into an equivalent statement about closed sets. You have to do that in circus sometimes. However, geometrically open and closed sets play different roles, different roles. So in our mind set, there is something about almost as if open sets are more important. But logically, there is no difference. You must see that. Let X tau be any topological space now. I am continuing with the definitions. A subset is called a neighborhood of a point. The point is also inside X of course. If there exists an open set U such that X is inside U and U is inside A. This is a definition of general neighborhood. Sometimes I am going to write NBD, this nomenclature for neighborhood. We should use the term open neighborhood that is O NBD when A itself is open. A is open and X belongs to A, then A will be an open neighborhood of X. So this is the terminology. It is fairly easy to see that a subset A is open. If filled only if it is the neighborhood of every point belonging to it. So I will let you think about this one. In the second part, I am not assuming open name just a neighborhood. Yet if it happens for every open set, every point, if you know it is neighborhood of every point inside A, then A must be open. And conversely, conversely obvious. If A is open, it is a neighborhood, it is actually open neighborhood. If you know enough set theory, you may derive the first axiom T from A U and F I. A U is arbitrary union and F I is finite intersection. Hence, putting this condition is somewhat redundant and some books do that. They do not put it at all. They just put only two axioms and others put it four axioms by cutting T itself into two. So basically axioms are important which are the only A U and F I. So we have induced, we have included it for the sake of clarity instead of being two thirds though logically only these two are sufficient. And by the way, there is a general understanding that a definition should be as short as it please, as it can be. No extraneous explanations, terms, etc should be there. This is the guiding principle of the definition. Okay. If you follow that, this T has no value. T should be removed because that is a consequence of A U and F I. Also not so obvious is the fact that intersection over an empty family is the whole sector. This gives you at least the beginner some problem. Okay. So I want you to think about it. This is a very important thing. If you proceed further and further in anthropology as well as in mathematics elsewhere, this becomes a necessity to understand this one. Okay. So make a beginning. While dealing with topological spaces, you often have to use these facts namely that a empty union is what? Empty union is empty set, empty intersection in the whole sector. So our topology was after all motivated by metric spaces. So there is no wonder if metric spaces give you a big source of topological spaces. How start with any metric space and this take this tau D that is topology. That is what the previous theorem says. 1, 2, 3 properties we have proved and those properties have become axioms, that is all. So x, tau D is a topological space associated with the metric D. So such a metric, such a topology, we will call a metric topology. That is coming from a metric. Sometimes we say it is matrizable instead of saying which metric it gives you. There may be several metrics giving the same topology. So it is just given matrizable topology. Obviously there is going to be more stringent condition on topology being a matrizable. They are some small class of topological spaces. Once you have created this definition, it is like a monster. It is no longer in our control. It will create lots and lots of topologies. So that is what I am telling here. If you have only metric topologies all the time, then there was no need to make this definition at all. The fact is that the great people who worked all these things, had bingers of topology, had foreseen the floodgates that this definition opens up. Our next step is to study a few examples of topological spaces which do not arise naturally as metric spaces. Topology is for metric spaces. But this we will do next time. Thank you.