 Hello, welcome to NPTEL NOC, an introductory course on point setupology part 2, module 28. This chapter consists of NETs and filters, so module 28 will be on NETs. Let me tell you a little bit background here, recall that by a sequence in a set x, we mean any function s, it domain the natural numbers into x. So, why it is so different from the set theory of functions, you know a sequence is just a function. The only specialty is that the natural numbers have a natural order in that, which is actually a total order which is actually more than that, it is actually what well ordered. So, that is the explanation that a sequence is more special than an ordinary function, it is a very special well ordered set natural numbers, which allows you to make so many other mathematical statements beginning with the principle of mathematical induction. So, there are many uses of sequences, but the one which is most relevant to us is the concept of convergence. The first thing that we notice here is the total order, we do not worry about the well order so much. The second thing is countability of N, that seems to play a major role. However, well ordering of a as I told you is not at all important for the convergence theory of sequences and whatever. The convergence property of a sequence is independent of its first few values to be very precise, that is why I think that the well ordering is not all that important. Based on these observations, we make the following definition, wherein we want to enlarge the scope of this convergence theory. So, the enlargement or the generalization comes in the domain of the sequences. So, naturally once the domain is changed, we do not want to call the old name, we want to change the name also. By a direction on a set D, we mean a partial order just to distinguish from the standard partial order on the real numbers and natural numbers. We are going to use this you know latex notation, you can just read as followed rather than you know anything else greater than less than or equal to is rather misleading terminology. We are not comparing any quantitative thing here but partial order is a partial order anyway. So, this is a partial order with the following property. When somebody says partial order, people already mean something, it does not matter. Actually, I will take a binary operation here that is just a meaning of that subset of D cross D satisfying the following property. A is always less than equal to A that is the reflexivity. Once you have partial order, I do not have to say this one but I am saying that because what I mean by here is clearly I want to state it here. A is less than equal to A for every A. A less than equal to B less than equal to C implies A is less than equal to C that is its transitivity. Then there is a third one which makes it a direction, what is a direction? So, this is not important given any two elements A B inside D that is a third one which sits over both of them or you may say which follows both of them. A is less than equal to C, B is less than equal to C. A and B are given, there must be such a C. If a relation like this satisfies A B C, I will call it a direction. A set together with a direction is called a directed set and is denoted by usually D comma break or D comma followed by D comma less than or equal to whichever one you want to read. Quite often as usual we will not mention this one when it is understood. We will say D is a directed set. Given any set X, now I am going to come to the sequence part. Now sequence is generalized as a word net. By a net in X we mean a function from directed set D to X that is all. Once again I want to have a word of caution about my terminology here. You see in partial order people have to have the anti-symmetry built in. In our case if we assume anti-symmetry no harm is done but the general definition of directed set does not make this assumption namely there is anti-symmetry. A may be less than or equal to B. B is also less than or equal to A yet A may not be equal to B that is allowed. So such a generalization is not of much importance for us. So if you do not want to bother you can assume anti-symmetry no problem because our examples are anti-symmetry. The general definition is necessary in what are called as directed systems in the category theory. Now let us have some examples. The set of natural numbers which was a motivating example is a directed set with the usual order. It has many more other properties those things we have you know sideline now. So a sequence is a net but now there will be many more nets than sequences. So we shall study them. First let us just concentrate on directed sets. An important example of directed set for us in topology is any local base Bx at a point x inside x, x is a topological space there is a local base right. With the usual inclusion of set inclusion relation defined by only the thing is I would take usual yes but it is reverse. So I would keep calling it reverse at inclusion. A is less than or equal to B now implies for us and implied by B is a subset of A. So people do use the notation instead of this prick they will use the reverse prick. There is absolutely no loss of generality at all. On the other hand there will be other things which will collapse with this one. If I use you will see the next example. For these examples and many more examples it would have been beneficial to denote A greater or equal to B if and only if A contains B right. That is alright but that this much of I should say that by spending this much of time I have removed any confusion here. In particular every local base is like this take the standard local base namely set of all neighborhoods which I will denote by this curly nx ok all neighborhoods of a point x inside the topology. It will depend upon the topology of course that is a directed set the same kind of reverse inclusion ok. We shall be using these directed sets in the sequel these are important for us. Some more examples an important example of directed set in topological spaces in the following. Start with any family C of closed subsets of x with finite intersection property. What is finite intersection property? Intersection of any finitely many members of C must be non-empty. In particular all members of C must be non-empty ok. Under the reverse ring inclusion this may not be a directed set. So we have opportunity to study this kind of classes. So we will not leave it like this but we will make it into directed set by consider the family D of all subsets which are intersections of finitely many members of C. Put all of them inside D ok. So you have enlarge the family C to this family D. All members of C are there already right because I can I can take just them intersection with itself is the same set all those are there. Two of them are you take intersection may not be there put that one also like that finitely many intersections should be put inside ok. Then D becomes a directed set under the reverse ring inclusion. Once you have two elements D1 and D2 which may not be comparable. You take D1 intersection D2 that will be smaller than both of them ok D1 intersection D2. So the directed condition 3 for direction is satisfied ok. Yet another example is the family of all open coverings of a given topological space x. With the relation ok I am going to define this one ok this relation denoting the refinement relation. What is the mean of refinement relation? Suppose u and v are open coverings u is less than or equal to v ok would imply that so here I have defined already. If u is u I belong to I and v is v j j belong to j are two family subsets of then we say v is a refinement of u and we write u less than or equal to v ok say refinement are coming for afterwards ok. Each member of v will be contained in a member of u. So there is a refinement function alpha from the indexing set of v namely j to the indexing set of u namely I such that v j is contained inside u of alpha j ok. So this is the refinement relation alright. Once you have two families like this you can take u i intersection v j where i runs over i and j runs over j this will be a common refinement. So that is why this is a directed set alright. So notice that here neither u is contained inside v nor v is contained inside u members of v are contained inside some members each member is contained some member. So that is the relation here right ok. Another interesting example of a directed set which is more or less the mother of all these you know theories is the Riemann integration theory comes in the Riemann integration theory. How you start with a bounded function on a closed interval and then you start cutting down the interval the partitions then you are not satisfied with that you take any two partitions you want to a refinement of both of them and that is precisely the notion of you know directed system ok. A refinement of a partition you know what it is because if it is a partition what is the refinement you put some extra points in between that is a refinement. So if you have two arbitrary partitions you can always get by you know interlacing them the points of one partition and another partition you get a refinement of both of them and so on alright. So for example suppose P is a partition like this we can define I am defining SF from this one the sum here associated with F at P and at P means corresponding to the partition P F is a function bounded function as A i minus A i minus 1 this is the length of the interval multiplied by the value of the function at the midpoint of the interval take the sum ok. So such things are studied and then you go on refining these partitions alright and what I want to just tell you I cannot go on doing Riemann theory here is Riemann integration theory can be formulated beneficially if you use the terminology of of directed sets directed sets and directed systems ok. So like this you can mention other things also from analysis which you may not be familiar with but I will tell you what it is it is used very much in advanced topology as well as in complex analysis it is called Runge's trick that is also similar only thing is this time you are dividing rectangles ok or domains inside C by rectangles smaller and smaller rectangles and so on. So that is famous known as Runge's trick you can use this to prove many things in complex analysis ok. So let us proceed with certain notions of sequences which we use in the convergence theory and try to modify them or adopt them for the case of nets. So here I am going to you know introduce subnets eventual subsets and so on. Start with any directed set take a subset of D and then you want to tell what kind of subset you are taking so there is qualification so it will be called eventual subset if there is an A inside D such that A is less than equal to B would imply B is inside D prime ok. So such thing is eventual so there is one point everything following that point that is one point everything following that point all those points must be inside D prime. So such thing is called eventual subset eventual subset inside now you can apply this one to nets also right sequences also are the ones which will determine the limit property of the sequence after all is what I have told you first few elements first few values of a sequence do not matter we are throwing away first few is precisely corresponds to this one now there is no first few and so on here. So you begin with some A and you are not worried about what happens before A only thing is everything after A must be inside D prime such thing is called eventual subset given a net S in X and a subset A of X ok SC say net means what S is function from D to X now we say S is eventually in A if S inverse A set of all points which come inside A under S that is S inverse A this is an eventual subset of the domain D of the function S ok so that is eventual set next we say net inside X where X is now a topological space see eventual set can be defined without reference to an topology that is the part of the definition of net now we are coming to convergence theory here so X must be a topological space we say S converges to a point X inside X if S is eventually in every neighborhood of X that is what is the meaning of this one in that case we say X is a limit point of S. So I will explain what is this one this just means that given a neighborhood U ok S inverse U must be an eventual set which is same thing as A given a neighborhood U there exist A inside D such that B is following A or A is followed by B implies S B is inside U ok so that means that the net converges to X in that case we say X is a limit of S ok now we immediately come to a theorem here which is a one point higher score than the sequences for a sequence you do not have such a theorem ok so what is it start with any topological space X it is Hausdorff if filled only if every net S in X has at most one limit it does not convert to two different points of X ok we know that in a Hausdorff space a sequence has this property but if every sequence has this property we also know that we cannot say that the space is Hausdorff the same thing here we are saying that if this happens for every net inside X then X must be Hausdorff the only part is similar to the case of sequences so I will leave it to you to figure it out as an exercise now I have come to if part so assume that X is not Hausdorff then I will prove that this property does not hold ok so that is same thing as proving property implies Hausdorff not Hausdorff implies not this property that is what so what is not not property means there exists some net S which converges to at least two points if I prove that then I have finished the proof of this theorem ok so suppose X is not Hausdorff then there are two points X1 not equal to X2 such that every neighborhood u1 of X u1 of X1 intersects every neighborhood u2 of X2 ok this is the negation of a Hausdorff now you look at the product set n X1 cross n X2 where n X size are neighborhood set of all neighborhoods of X1 you see if this one is a is a directed set this also directed set but what is the directing on the product I will have defined ok there are many ways of defining it ok so define this direction on D as follows u1 comma u2 they are members of n X1 cross n X2 right is less than or equal to v1 comma v2 if and only if if and only if this is a strict order u1 is less than or equal to v1 and u2 is less than or equal to v2 so u1 contains v1 and u2 contains v2 ok both of them should hold verify that this is a direction on D ok is it easy you can do that right why now define S from D to X by the rule S of any ordered pair u1 u2 is an element of the intersection ok this is intersection intersection of any two members one from here one from here it is non-empty is the assumption for X1 and X2 here ok every neighborhood u1 intersect every neighborhood so your intersection u2 is non-empty so I can pick up a point ok to define this function you have to say we are using an axiom of choice here ok we are using axiom of choice all the time anyway so let S then S is a net alright once D is a directed set function is defined S is a net this net I want to claim converges to both X1 and X2 ok it is very easy to see take any neighborhood ui of Xi i equal to 192 we can select u1 comma u2 belonging to D which satisfies the property that if v1 v2 follows u1 u2 that means v1 is contained inside u1 and v2 is contained inside u2 then by the construction S of v1 v2 is an element of v1 intersection v2 which will be both inside ui i equal to 192 u1 and u2 are arbitrary I have found out S of v1 v2 so it is inside ui for all of them which follow u and u2 so S converges to both X1 as well as X2 ok in one go we have proved both of them alright in any case I want to recall this example which you must have seen in part one already take an uncountable set R like the real numbers itself with the co-countable topology COC co-countable ok remember co-countable topology means what a set is open if and only whose complement is countable or of course M or it could be the whole space the empty set is also allowed now take a sequence actually S from N to R be a sequence which is convergent to a point R inside this this space ok put A equal to S of N minus R S of N is a countable set that is all throw away R from it that is also countable set then A is a countable set therefore R minus A is open but little R belongs to this R minus A right because A does not contain it so R minus A is a neighborhood of R now SN converges to R ok start with a sequence converging to R what I said so this is a short notation for this S of N is SN SN converges there exists N0 such that N is greater than or equal to N0 implies SN belongs to R minus A which is a neighborhood of little R but SN intersection R minus A is just singlet and R because everything other thing has been thrown away here is R minus A only R will survive that means SN is equal to R N greater than or equal to N0 what is what is this mean this means that the sequence is eventually a constant right thus we have proved that every convergent sequence in R with co-countable topologies eventually a constant in particular a sequence in R we have a co-countable it can have at most one limit right it is essentially constant in the constant is the only limit right so the property is satisfied however we also know that the co-countable topology on this uncountable set is not hostile any no any two non-empty open sets will intersect okay so this is one little small surprise for you or justification for doing something general than sequences okay so we will have many such things so next one yet another aspect in which nets fare better than sequences in the following result recall that we have defined a space to be sequential if its topology can be determined by convergent sequences in it right so sequences have this property while studying you know metric space and so on in this case in the case of nets if you try to do that you will get all the topological spaces so it washes out the whole thing okay so that is what I want to say that means what the sequences will determine the topology completely how so it is what here then this theorem that x be any topological space and a be a subset of x okay take non-empty subset because a is empty set then there won't be any statement then a is open if and only if every net s in x which converges to a point in a is eventually inside a okay so that is characterization of an open subset other than non-empty set right other than empty set characterizing all open sets means its topology completely determined by the sequence behavior of the sequences the behavior of the next next so this is a theorem which is not at all difficult to prove but the definition of convergence of a net the only if part is clear eventually should be inside that part is clear we need to prove the if part suppose a is not an open set okay this means that there exist a point x inside a such that a does not contain any neighborhood of x because if it contains neighborhood usually in the interior right so every point is in the interior the a is open so that is at least one point which satisfies this property namely no neighborhood of this point x is contained inside a all right now look at any local base bx you can take the whole of nx if you like but just local base is enough sometimes you can verify this to only local basis that is why I am using local base all local base bx at x okay as in example which we have studied all day local basis are directed set for each b inside bx choose sb to be a point inside b intersection complement of c complement of a because a complement is you know this neighborhood b intersection a complement is non-empty that is what we know it is not contained inside a right so b intersection a complement is non-empty I can choose a point once again this is this is the definition of s use is axiom of choice but it follows just as before that s converges to x but it is never inside a no point of no point of this sequence is inside a forget about eventuality all right so we have seen that convergence of nets determines the topology so it is a powerful suddenly okay so we will see its power a few more properties of this convergence theory we will study once again there are topological space is x which subsets a such that every sequence which converges to a converges to a point in a is eventually in a yet a is not open okay I will not give you specific example here but it is already there whatever you have seen today so just find it out let here is an exercise let D prime be an eventual subset of a directed set show that D prime and the same partial ordering the same direction restricted to the D prime okay is a direction on D prime so this becomes a directed set so this is something non-trivial have to show it is not difficult further take any sequence any net on D sorry the net s from D to x inside x suppose you take a see first part was only about the eventual subset inside directed now take a net okay defined on D inside x suppose converges to x belong to x then the sub this is restricted the sequence restricted net that converges and conversely which is defined on F so this was not difficult figure it out once you do that you will be familiar with the definition of convergence definition of eventual set and so on okay so let us stop here next time we will study little more properties of next thank you