 Welcome to module 7. As promised last time, let us now take up one by one a few examples of topological space is directly which may or may not arise from a metric space. We shall begin with a few examples of topologies which do not arise naturally as metric topologies. So, that may be the end of today's lecture. Just some more examples. Let X be any set. Then we take a subfamily namely consisting of empty set and X. We cannot do more stringently than that because the first one, first axiom says empty set and X must be there. So, put that, do not do anything more. That is it. The other two conditions are automatically satisfied. What is the subfamily of these? Either it is just empty or just X or both X and empty. What is the union? Either if you have empty set, union will be empty set. If you take X also then union will be just X. If you take the intersection, it is always empty. If you take only singleton X as the set, then intersection is X again. So, AU and Fi are automatically satisfied by this one. Similarly, there is another, well, you may say these are in disinteresting. In some sense, it is disinteresting. In some sense, it is extremely interesting also. So, take other way around namely put all elements of Px, all subsets of X. So, that we are going to denote by this curly D here. This is curly I for indiscreet space. This curly D for discrete space. These are the names. And checking that this curly D is a topology is very straightforward because we have made no restrictions at all. Every member of the power set is there. The power set automatically decides all these properties. So, what is happening here is this I, the indiscreet space, this topology is the smallest topology. You take any topology, it will contain this I. Similarly, this D is the largest topology. Every topology is subset of this by definition because D is Px. There is no more, there is nothing bigger than Px on X. So, the least one and the biggest one, that is what we have observed. Now, these things did not actually come from metric space already. We have created it without reference to metric space. But maybe they are matrizable. That is a different problem altogether. So, let us examine that also later on. Let me give you another important example here. By the way, this fellow is also the very important person. He has done a lot of work not only in topology, but in number theory, geometry, all sort of things. So, he was a Polish mathematician, Sir Pinsky. Maybe you have to pronounce it as Sir Pinsky. An interesting example from theoretical point of view is the so called Sir Pinsky space. Here we take X to be a set with two elements. So, two elements set usually you can denote it by 0, 1 or minus 1 plus 1 depending upon what kind of algebra you want to do. Okay, here we are not doing to do algebra. So, you choose whatever you can trace, you can take any A, comma B also. I do not care. Okay, take any two elements set and only proper open set to be singleton 0. When I mean proper open set, I mean the one which is not in the list 1. Namely, the whole set is always there and empty set is there. They are not called proper open sets. They are just open set everywhere they are there. So, proper means not equal to empty set, not equal to whole set. In between them there is only two of them namely singleton 0 and singleton 1, but do not take singleton 1. They take singleton 0. Okay, only one of them you take, that is the meaning of it. That will be a topology. Let us verify that, let us denote it by as verification that this topology is very straightforward. Okay, why I am putting this one? One of the properties which you can verify later on because right now we have not done enough terminology here. In the following, the space, the space has a wonderful property that take any other topological space Y. You will know completely if you know all continuous functions from Y to S. So, right now you do not know what the meaning of continuous function from a topological space to another topological space. So, right now you just take it as granted, but once you know what is a continuous function, you take it as an exercise always. Okay, knowing all continuous functions the same thing as knowing Y means what you should be able to describe what are all open sets here or what are all the closets here. That is the meaning of knowing the topology. Okay, so for that reason, there is one reason that there are other things, this will serve as a lot of illustrating examples or what you may call as contra-examples. Okay, so there is a method, this Sierpinski space construction is a method that I am going to describe now. The general construction and then there are variants also, one part I will tell you is very useful in giving contra-examples. Start with any topological space extra. Okay, take this SX. What I have put this notation, I mean because I want to celebrate this guy's name Sierpinski. So, this is Sierpinski operator. So, what does it do? It pushed an extra point inside X. So, like what we have done in the extended complex plane like that. Okay, X disjoint in an extra point and the topology tau prime. Okay, where underlying space is a disjoint unit of X with an extra element and this tau prime topology consists of all members of tau, they are there. Anything which is subset of X and open in X, open in tau, belong into tau is already in tau prime. So, all members of tau are there. Together it is one extra element namely X union star itself. Thus the only open set containing the star is the whole space. So, that is the property. So, this star which has the property that look at any open set containing it is the whole space. Such a point is called Sierpinski point. Okay, such a point is called Sierpinski point and such a space here what I have called SX is generalized Sierpinski space. So, starting with the X, if you obtain this SX, you can call it as Sierpinski equation. For example, in this example, what is the Sierpinski point? It is 1. 1 is a Sierpinski point. Okay, because take look at any open set containing 1, it will be the whole space. For 0, there is an extra 1. 0, singleton 0 is an open set. So, 1 is a Sierpinski point here. Okay, another interesting topology which is quite celebrated, which is quite central than Sierpinski topology and so on. For this, I have to start with an infinite set. Then this co-f corresponds to you know complement of finite. So, that notation to be the collection of all subsets of subsets A of X such that the complement is finite. Of course, A equal to empty I have to take here because complement of A is not finite, but still I have to allow empty set also just to take care of the X empty. It is not hard to see that this is a topology on X. As soon as X minus A is finite for 1, to take unions of several of them, X minus a union will be also finite. Even one of them is finite, so that is very easy to see. If A1 and A2 are there, X minus A1 is finite and X minus A2 is finite. X minus A1 intersection A2 will be also finite because it may be union of theta by De Morgan law. So, you can easily verify that this is a topology. This topology is called co-finite. That is why I am writing off co-finite topology. One of the statements about that is that if you take all neighborhoods of a given point, take any point, this is going to be true for all the points. Take intersection of all neighborhoods. It will be precisely equal to X. So, this is the characterization of this space. Take it as an exercise. Think about it. Later on, we will give you several characterization of this topological space. Again and again, this will be used. Different way of looking at this same space. Somewhat similar to the above example, but not so important is the following. Instead of taking just an infinite set, now you take an uncountable set. That this now of C here, COC, you know the set of all A such that either A is empty or X minus A is countable. Countable includes finite also, but it could be countably infinite. That case also there. So, this will be larger. So, that is why I have started with an uncountable set and put this condition. Once again, verification that this is a topology is similar to, I mean you have to use De Morgan's law. This will be called co-countable. That is why COC, co-countable topology. So, to understand what is going on, you can compare it with the earlier example number 4. Now, I come to more familiar things. These things were just to tell you that metric is not all sacrosanct. There are many other ways of getting topology and interesting topologies. So, I come back to the real numbers. I again look at the metric. Okay, dx, y is here, x minus y. And it gives you topology and that topology we are calling the usual topology or Euclidean topology. So, it is denoted by you by all, almost all authors, currently you for the usual topology or you may call it as Euclidean topology. It can be described slightly different way that is my object now. Instead of looking d as a metric, now I am looking at the order in r, x less than y. So, in the order in y, there is a total order in x. Consider all the open intervals. When I say intervals that refers to the order not to the distance. The interval means what? All points between, lying between any two given points. So, I know what, I hope you know what is the definition of interval. Then take the collection of all subsets which are unions of members of this union of intervals, open intervals. Of course, empty set is also allowed. So, this is similar construction here is similar to what we did for tau d. There we took the balls with the metric. With the real numbers, you do not have to refer to the metric but you can use the order. If you take for example, interval between 8 to b, it can be also thought of as an open ball is center a plus b by 2 and radius equal to b minus a by 2. So, in some sense, I am cheating but no. You see, I am never mentioning the metric here. I am just looking that the order here, which order is not available if you go to r2, r3 and so on. So, it is available only inside r. So, that is why I am using it. It is not even available for complex numbers you see. So, that is important here. So, you can describe this topology as a open set is what? It is union of open intervals. Then it is not very difficult to prove that the third property with intersection of two intervals is either empty or it is again an interval. Open interval intersection is open interval that you can prove very easily. That is a part of the proof here to show that this forms a topology. Now, I come to why I will introduce this one. This idea of order in R can be generalized quite usefully. Recall that by a partial order on x, we mean a binary relation on x which satisfies reflexivity, transitivity, x less than y, y less than z implies x is less than z. And anti-symmetry, if x is less than or equal to y, y is less than or equal to x, less than or equal to it should be implies x is equal to y. In addition, if this operation, this partial order has the property that any two elements in x are comparable. So, I am putting extra condition here. That means there is a total order. So, I am recalling this. I hope you already know such a thing. If you do not know right now, you can learn it. So, total order or linear order. We also have the short notation x less than y just implies x is less than or equal to y that was the relation, but x is not equal to y. Now, I have deliberately denoted this one by this Curly's symbol because right now, x is not the real number. So, not to confuse if the standard less than or equal to inside r, I have denoted by this one, this prex symbol. Now, start with a total ordered set x now, x sorry, x prex. We can then take all open intervals in x. So, let me redefine what are the open intervals here now carefully. Given x and y, you can look at this notation x comma y bracket. This is also unfortunately the standard notation for ordered tuple. Here it is an interval, all point z inside x which lie between x and y, x less than z less than y. Okay. So, again here there is a typo. I want to write this Curly here, Curly thing because that is the notation here. Never mind. So, x infinity is all y such that y is less than x is less than y. See this infinity is not a symbol which is x comma infinity is a symbol. Infinity itself does not make sense here. It is not here. This is neither in x nor something like in the case of r, something you know, it all converges to n, n1, n2, n3, n going to infinity converges. That is not the point here. There is no convergence or anything like this x comma infinity is an interval. How to write this one? This is an interval. All points y which are less than x. Sorry. All points y which are less than x. Okay. Similarly, minus infinity x is all points y which are less than x. Take all these, they will form a what? They will form a topology. Where is the empty set? You can take x equal to y, then this will be an empty set. Okay. There is no z between them, strictly between them. This is the point here. Okay. So, that is a topology and that topology is called order topology. So, what you have to check here? The simplest check you have to prove is that intersection of any two open interval is again an open interval if it is non-empty. So, you do not use real, the topology of the real set all only use this one. That is the whole idea. So, it can prove. If you are given a topology on a given set, that may arise several metrics out of several metrics or it may not arise out of any option. If you take a finite set, for example, then every metric on x gives you the discrete topology which I denoted by in a curly day. On any set x, the topology induced by the Gritschitz metric is again discrete. On the other hand, we shall soon see that there are many topologies which do not arise from any metric. Of course, simple example we have already namely you take a set with exactly two points but take the indiscreet topology. Then it cannot be out of any metric because as soon as you put a metric which induces the topology, the topology will be discrete topology. On a two points set, discrete and indiscreet are different already. In fact, it is true that you take any indiscreet topology on any number of points. It cannot come out of a metric. That may need some more seeing once we understand what happens to metric topologies in general. So, having said all this, we still want to stick to a lot of ideas and lot of results from metric spaces. Why? Because the central theme in topology is always parallel to the results that we have proved that we get from metric spaces. So, we should not forget metric spaces or discard them. Now, let me give you one of the important examples. It looks as if I am doing algebra now but those algebras are very, very, very trivial algebra. The algebraic structure that we have on the complex numbers or real numbers, the field k. In the field k, you can add, you can multiply, addition and multiply have interrelation, associativity, commutativity, law of distributivity and so on. These are the things which make a formal definition of a group and a ring and so on. So, you do not have to worry about too much about those structures. So, whatever I am going to do, you can take that as definitions. If I introduce new notation, new term, you can take that as definition. It is that much accurate there. So, having said that, take any set and look at all the functions from that set to k. Now, point wise addition, point wise multiplication is the key. Take two functions fx, f and g. You define f plus g by taking their values and then adding them. f plus g operating upon x is fx plus gx. Similarly, alpha times f operating upon x is alpha times fx. So, this all day makes the set of all functions from x to k itself a vector space. The vector space structure actually comes from k. So, that is a vector space structure. Moreover, I can also multiply two functions because I can multiply two numbers. fg of x is fx into gx. So, that multiplication will have again the same kind of relation with the addition namely it is distributive. So, what one can say is that bilinear with respect to the vector space structure f of g plus h is fg plus fh and you can pull out alphas wherever you like. Alpha f of g is the same thing as f of alpha g. So, that is the vector space structure. Linear maps, that is what it means. Bilinear means what? Linear in both the variables. Not only that, even on the other side also. g plus h of f is the same thing as gf plus hf which is same thing as fg plus fh and so on. That is because multiplication is commutative also. In particular any such structure is called a commutative ring. So, this capital F, this currently F which is set of all functions becomes commutative ring. It is a commutative ring, it is a vector space. Such a thing is called an algebra. So, this algebra has another important property namely the constant function 1 plays the role of multiplicative identity. f into 1, operative on x, fx into 1 operative upon x is 1. So, fx into 1 is fx. So, that is why this is called a commutative algebra over k with an identity element. If you want to know other examples, the most famous examples, most useful examples are polynomial algebras. Take complex numbers as coefficients, take one variable polynomials, they have all these properties. Take real coefficient, take one variable, they are all these properties. More generally, you can take n variables and take k x1, x2, xn which denotes the set of all polynomials in the variable x1, x2, xn. Since it becomes a what is called a polynomial algebra, it is exactly similar property. Actually, this polynomial algebra is a subalgebra of our this great capital F. How do you see that? The special case namely take the set x as just k power n. So, they are all functions on k power n. Do not take all functions, only take polynomial functions that is the subalgebra. Now, I am going to give you another important subalgebra here. Namely, I am writing it as b or curly b maybe. This is b corresponds to for bounded functions. Only take bounded functions. Sum of two bounded functions is bounded. The scalar multiple of a bounded function is bounded. Product of two bounded functions is bounded. So, this is what easy to see. What is the meaning of bounded? As x varies, modus of fx is bounded by a single number say m. For all x, this should happen. But then f is bounded. For each f, you may have different m. If f is bounded, g is bounded, then you can find a common bound for f plus g as well as fg, as well as an alpha times g and so on. So, set of all bounded functions forms a subalgebra. Subspace as a vector space, as a ring, everything it is subalgebra. On this subspace, we can do some geometry. We can put a norm now. So, the norm is if you do not want rigorous, again a rigorous notation, a short notation is just norm. But it is the infinite norm. This notation is used for supremum norm. What you will do? Take modus of fx as x range over x. Take the supremum of that set. It is a bounded set. The supremum means a finite number. Then the verification that it is a norm is exactly similar to what we have done for other cases of soap. Only triangle inequality takes a little bit of time. Unless you spend that time yourself, you won't know what is going on. That much of time you have to give on your own. So, you verify that this is actually a norm. Moreover, there is one extra property here namely norm of fg, the product is less than or equal to norm of fg to norm of g. This is not a part of the definition of the norm. But the norm we have, norm of f is 0 if it is only if f is 0. Alpha times f, norm will be mod alpha times norm f and triangle inequality. But this is an extra thing because there is an extra structure. You can multiply two functions here fg. It is norm. It is less than or equal to norm of fg to norm of g. This property of the norm, which is in some sense it is respecting the modification. So, that makes it a normed algebra. The norm and the algebra structure are not just on their own. They have some relation and that is that this is relation. Then it is a normed algebra. Once you have a norm, of course, as usual you take the distance function corresponding to that. Namely distance between fg, f and g is norm of f minus g. So, that will become a metric space. That is what I meant by geometry here. Now you can talk about distance between two functions and so on. Now you can talk about convergence. Convergence, whenever convergence you have to look at the metric with respect to that metric. You can talk about convergence of these functions point wise because f of you know f of xn, fn of x and fn is a sequence at a point. You get a sequence of real numbers or real sequence of complex numbers. You can take the convergence of that. That convergence will be point wise convergence whereas this convergence here with respect to sup norm always gives you what is known as uniform convergence. This uniform convergence is exactly the same thing as what you have done in your calculus course. If we have not done it, we will whenever we use it more seriously we will tell you what it is. But this is not part of the topology for this actually. Analysis you must have learnt. Algebra b has another interesting sub-algebra there only when you take x as a metric space or a topological space and so on. You take x as a metric space. Remember b consists of all bounded functions. But now I want to put extra condition. You see set of all those elements in b which are continuous also. Now continuity makes sense because here I have a metric space and here I have another metric space. You can think of this as metric space always when you do not mention anything. This should be taken as a metric space with the usual topology namely the modulus. With respect to that you look at all continuous functions. We have as well as natural definition also or you can use sequential continuity definition also look at all continuous functions. Then you can show that exactly same way as in your calculus courses if f is continuous, j is continuous, f plus j is continuous, alpha times f is continuous, f into g is also continuous. So what does that mean? That means that this cxk, this is a sub-algebra like bxk sub-algebra of f, cxk is also sub-algebra of bx now. Not only that, one of the most important property of both b and c is that they are complete with respect to this now. Now what is the meaning of completeness? Similar to what you have done in other cases namely in the case of r and c and so on. Every Cauchy sequence is convergent. That part makes it a banak algebra otherwise you would have called it only a normed algebra. These two are b and c, they are actually banak algebra, they are complete also. So this part we will come back to it, we will study these things more thoroughly later on. So this is an important example, we may come back to it again and again. Let me now go back to giving you more examples of topologies. Once again look into k with the modulus function. The modulus function I told you gives you the usual topology, the so-called usual topology or Euclidean topology. This topology now can be expressed in a different way when k is r, not for complex numbers. When you take only the real numbers, you can describe it in a different way namely look at the order there. Each number can be compared with another number, there is an order there. So using that order you can talk about intervals. The open interval x, y means all those points t lying between x and y strictly and so on. So you have open intervals. Now you take collection of all open intervals. Then just like we have done in the metric space in general way, what you do? Take unions of all open intervals. I mean not all open intervals, only open intervals, any family of open intervals. Take all such possibilities, open intervals, two of them, three of them, infinitely many of them also. You can take unions of such things. Each of them, you put them together in one single collection that will form a topology. What is that topology? It is precisely the same as dxy, given by dxy, namely open bars, closed bars, open bars or open intervals here. But now I do not want to refer to the metric, but I just want to talk about intervals. That is the only, it is the same coin, I am looking at a different side that is all. So that collection is exactly the usual topology. But why bother about looking at a different way? To look at that, that is the whole idea. Now you can just forget about all the additive structures, particular structures and so on in the real number. Just look at the order itself. That is enough for me to decide the topology there. Therefore I would like to look at other structures which may have just the order only. So that is what I am going to do now. The idea of the order in R can be generalized and usefully, it will be useful something. It is not just for generalization, it is for generalization sake. We recall that by a partial order. If we have not done what is a partial order, I am going to recall it here. So we mean a binary operation on x which satisfies reflexivity, namely x is always related to x. Let us read this one as preq just to distinguish it from the usual less than or equal to. x is preq x for every x. x preq y, y preq z implies x preq z. This is transitivity, anti-symmetry. If x is smaller than y and y is smaller than x, then x must be equal to y. So same thing here. x preq y, y preq x implies and implied by x is equal to y. So these three properties define a partial order. In addition, there is one extra property I am going to assume, namely given any two elements, given any two elements in x, that is y belong to x, they must be comparable. What is the meaning of this? x is less than or equal to y or y is less than or equal to x should hold. Then it is called total order or a linear order. So this property is also there with real numbers. So I am going to stick to only these much properties. Later on I may put more and more properties. Let us just take a totally order set. Let us have one more notation. Namely, when you write x strictly less than y, what is the meaning of this? x is related to y but x is not equal to y. So that is the meaning of this one. x is strictly less than y means x is preq y here, x is not equal to y. So this is a notation. With this notation, let us do something now, some meaningful things. Start with a total order. Then you define intervals just the way you define it for real numbers. But these are not numbers, no need. x open x comma y is all point z in x which are strictly between x and y. x preq y preq z but x not equal to z and z not equal to y. That is all. Similarly, you can define the open ray x to infinity as all points y such that x is less than y which means y is bigger than x. Similarly, all minus input x all points y which are y is less than x. So I have defined the intervals here. Now you take collection of all intervals and unions of all them. I am looking at all subsets of x which are unions of open intervals along with empty set of course. I do not have to say that because if you take x equal to y the open interval x equal when x is y, xx that is empty. There is nothing in between. So that is logic but let us just make it clear that empty set is also there and the whole set x is also there. So let us make this thing. Then it will be a topology. Now you have to verify it. You cannot rely on just your intuition or it may have done it for real numbers. What is the property that you use for the real number? That property should be here in the just order topology. If you use something else that will be wrong. For example, suppose I stop just with the partial order. I do not assume total order and I do the same thing. Intervals can be defined and I can take this one. It will not be a topology. Do you understand my point? The total order is necessary to assure that this collection becomes an order, becomes a topology. So take a minute, verify this one by yourself. Order topology has to be ensured. What you have to do? Just like in the case of open balls, you will have to check that intersection of two open intervals if it is non-empty must be again a union of intervals. If it is an interval it is fine. That is stronger. You have to show the union. That is what you have to show. Then only the union is coming but the third property finite intersection you have to verify. So let us carry on. So once more one more remark I can make. If you assume little more properties for this totally ordered set, that will be reflected in the topology and vice versa. So you may get more and more interesting things when you put more hypothesis on that. So this example will be again met at the end of the course or maybe at some other time but it will keep coming again. Just to emphasize this one, if I put more properties, then it will have more interesting things happening. Let us just put one example at a stake. Namely, I start with an order topology, order relation, totally order set and order topology. I have taken extra, I have fixed it. Now assume one more property of this order. Namely given any x, y, let us say x is less than y. Then there exists a third such that x is less than that less than y. So in between any two elements, there must be a third element. See the third element it is different from both x and y. I am not talking about, if x and y are real numbers, I can take x plus y by 2. That is not allowed here. But I am saying that this property is true. Let us assume this property. One can give a name but let us not bother about the name right now. So given any two elements alpha and beta belonging to x, I want to say that there will be an injective function from set of all rational numbers to this interval alpha beta inside x. So the whole of q is sitting inside alpha beta through this map, this injective map such that this eta is order preserving. So what is the meaning of order preserving? There is an order here, there is an order here. So it must be preserving and it must be injective. What does it mean? It is strictly, it is like a strictly monotonic function. r as than s in q implies eta r is spread eta s. So this is a claim now. I want to show this one. Just wonderful thing, just imagine what is the meaning of this one. This means that whenever it is such a thing, just one element is there between any two elements. This is hypothesis. From this one you can conclude that inside every non-empty interval you can find representatives like entire rational numbers. The whole of rational numbers will be there inside that. Yes, one one correspondents and it is order preserving. So such a wonderful thing will come out of that one. It is not very difficult but the proof is educative. It can be the proof itself, the idea can be used elsewhere. That is why I am doing it. The construction of eta is very straightforward. For all, first of all let us start with an enumeration. Enumeration means what? Labeling of the all the rational numbers. Rational numbers are countable. So I can put q equal to r naught, r1, rA. I am not saying now that this is smaller than this one. That is not possible. It is not in our total order. So some enumeration of q, we start with that. By the hypothesis, there exists x naught belonging to x such that alpha is less than x naught less than beta. Because alpha and beta are given, any two numbers are given there. Any two elements are given there inside x and the excess these properties are between any two elements. There is a third element. So take that one. We start mapping this r naught to x naught now. So eta r naught is x naught. Inductively, so the definition of eta is complicated, inductively. Inductively, suppose we have defined eta up to n, r naught, r1, etcetera, rn. So n could be 0, just 1 we have defined. So that whatever you have defined, that is already order preserving. Now I am looking at extending this function to rn plus 1. So I am going to define eta of rn plus 1. So rn plus 1, look at its status inside q. That q is ordered already, totally ordered. So rn plus 1, look at r naught, r1, rn. It will be somewhere in between. So where exactly it is, that is what you have to do. So I am making three cases here. Namely, first of all, there will be precisely two points a, b belonging to r naught, r1, rn such that rn plus 1 sits between them. a less than rn plus 1, less than b. a and b are already elements of r naught to rn. So somewhere it is there. The second condition is it is not inside anywhere here. It is bigger than, the rn plus 1 is smaller than all the ris. The third condition is r i is bigger than all the, rn plus 1 is bigger than all the ris. So these, out of these three, one of them has to happen. So accordingly, what we will do, select xn plus 1 belonging to x such that in the first case, select it between eta a is already defined and eta b must be bigger than eta a because eta is ordered preserving. These two are defined already. So pick up one element between them. See xn plus 1 has to be picked up by us now and this xn plus 1 is going to be eta of rn plus 1. How do I pick up this one between these because rn plus 1 is between these two. If the second condition happens, what is this one? Second condition, rn plus 1 is smaller than all the ris. Therefore, look at alpha. It is smaller than all the ris. Therefore, it is smaller than the minimum of them. So between them, there will be xn plus 1. So pick up one element here. In the third case, who do other way around? Namely, beta is bigger than all the, all these elements. Therefore, it is bigger than a maximum. So between the maximum of all these ris, you pick up not eta r, you pick up xn plus 1 between them and beta. Define r eta of rn plus 1 equal to xn plus 1. That is all. So let us stop here. Next time we will continue with more examples and better themes. Thank you.