 Hello, welcome to NPTEL NOC, an introductory course on Point Set Apology Part 2. So, today we will continue our study of NETS, module 29, Co-Final Families, Subnets, etc. So, let us begin with the definition. A subset D prime of a directed set D is said to be co-final. If for each A inside D, you must have a B inside D prime such that this B follows A. If you have a net D to X that will be called frequently inside a subset A of X if S inverse of A is co-final subset of D. So, this is the same thing as saying that given S inverse of A must be co-final subset. So, all this B, so whatever B you get, B should be inside this S of B must be inside A. So, take this as D prime, given A inside D, you have to get a B inside D prime such that S of B, B is bigger than B, B follows A and S of B must be inside A, that is the meaning of S inverse of A is co-final. S inverse of A is co-final, same thing as now S is frequently inside A, which is somewhat weaker than eventually inside A. Eventually a constant function is much more stronger than having just a subsequence being there. So, this is somewhat like let me say there will be a subsequence, but we will see that there is a notion of subsequence, there is a different notion here for subnets. So, it is easily seen that eventual subset is co-final, but converse is not true that I have indicated even you know you can have examples of sequences themselves wherein this is not true. So, another remark is a co-final subset of this Nx, remember Nx is a directed set in an interpolical space, it is nothing but a local base, remember that definition given any open neighborhood of S that is given a member of this one, there exist a member here which follows that, this is the same thing as there exist a member in the local base which is contained inside the given member here. So, that was the definition of local base right. So, every local base is nothing but a co-final subset of this Nx. Therefore, they will do the job of this Nx quite often, the following lemma is obvious. So, that is precisely what I mean for saying that the co-final families take care of the convergence properties of the original thing, every co-final subset F of a directed set is again a directed set with the same direction restricted to the subset. Moreover, if S from D to X converges to X inside X then sort of S restricted to F. So, that is why they do the job is what I said, but they may not do everything that is the original thing can do, if you took all of them then they will do the job that is whole idea. A subsequence of a sequence of a sequences is a co-final family of a treated as a net, a subsequence can be thought of as a net of a net where sequence itself is thought of as a net. So, both of them you take the net then a subsequence will be a co-final family in this sense. Co-final world is used even within sequences also anyway. One more definition, now we are coming closer and closer to the convergence property, take any net in a topological space a point is called a cluster point of S, earlier we defined limit point, limit of a set. Now cluster point of S if and only if S is frequently in every neighborhood of X which is same thing as saying that S inverse of U is a co-final subset of D for every U where U is a neighborhood of X which is same thing as say given A inside D there must be a B inside D such that A is less than equal to B and SB is inside U. So, this is the notion of cluster point just like subsequence converging. So, co-final thing converging we will call it as cluster point. Now here is a lemma which relates the property of being cluster point take any net suppose there exist a co-final subset F of D such that S restricted to F as a net converges to F then X is a cluster point. One such co-final family is enough. S restricted to F converges to F implies given a neighborhood U of X there exist X belongs to F such that F belongs to B A is less than equal to B implies SB is inside U. So, this is the meaning of S restricted to F converges to F. To show that X is a cluster point of S now given any P inside D choose first choose Q such that P less than equal to Q and A is less than equal to Q you have an A and a P right P is there. So, take one which is bigger than both of them which follows both of them that shows that S is frequently inside S. Hence we are done. See as soon as A is less than equal to Q S of Q will be inside it right. So, what is the conclusion here the cluster point is generally a generalization of a cluster point of a sequence in analysis. There we have a strong theorem namely a point is a cluster point of a sequence if and only if there exist a subsequence which converges to it. This leads us to think about a notion of a subnet of a subnet like a subsequence. First of all let us recall the correct definition of a subsequence sometimes some book do not give you so I am going to give you that one. T from N to X of a sequence S from N to X namely we must have order preserving function P from N to N such that this T is S composite P ok it is like a re parameterization right. So, order preserving map one way ok then this T is a subsequence of S experience tells us that a simple minded generalization like this replacing N by any two directed sets would not be enough ok. See all sequences are the domain of same N, but when you take all nets the domains keep changing here right. So, if you just say there is an order preserving map from one to the other even that may not be enough. So, you have to be cautious here. So, with sufficient experience we have come across this definition there is a scope for improving or making it more complicated or whatever I would like to say all these things are not hardened first rules right. So, you are free to think of doing something different also. So, here is a final I say final definition as far as the existing theory is concerned. Start with a directed set and a net inside X by a sub net of S we mean another net from another right another directed set E comma this brick prime such that there exist a satiristic function P from E to D just a function satiristic function E to D such that our T is nothing, but S composite P this is the first thing. Now, in the definition of a subsequence we have order preserving relation here order preserving map here. So, that is replaced by a weaker condition here S S 2 this S is for sub net S 2 S 2 says the following for every D inside E you must have an E inside E such that for all E prime which follows E E prime E less than or equal to E prime D must be followed by P E prime. So, P E prime should come after D. So, that means the property of P and take such a P take T equal to S composite P all such T will be called sub net of S. So, this is reflecting cofinality here of course, you have to bring P here if P is where an inclusion map then you could have been just like a cofinal map if E E prime is a sub sub of D D prime then it would have been the same thing, but we are now we want to prove any any two sets. So, they are related by a function P which has this property. So, this becomes a far superior definition in date let us see whether it works or not if you make a too weak a definition then it may not be good enough that is right. The first condition says as in the case of subsequence this is just this is the definition of of like a subsequence, but we have given up on the order preserving requirement of P instead of that the second condition is something like if E prime goes to infinity P E prime goes to infinity this is like that you see. So, E primes are larger than E then P E primes are larger than D that is like epsilon delta definition of function going to infinity if you if you do correctly. So, it is similar to that ok. So, analysis is the guide for all these things after all what happens there the experience there that is all. Notice that every subsequence of a sequence sequences is a subnet as a if a as a sort of a subnet also we can easily get examples of subnets of a sequence which are not subsequences ok very easy to get just obliterate one one element. So, that is not order preserving it has no effect on the rest of them it will be a subnet, but it will not be a subsequence that is all thus this definition of subnet is a liberal generalization of a subsequence ok. So, one more remark here one is tempted to compare the two concepts a co-final family and a subnet indeed if F is a co-final family in D comma whatever of partial order direction then with the restricted order it is a directed set right. Also given any net S from D to X taking P as the inclusion map it follows that S strictly to F is S composite F is a subnet. So, that is what I have told you that co-final families do give you subnets. Thus ok we see that the concept of a subnet generalizes the concept of restricting a net to a co-final family as well ok. So, we can now strengthen the earlier lemma that we had we had proved namely now it if filled only if ok in terms of cluster point take any net in a topological space then X belongs to X is a cluster point of this net if filled only if there exist a subnet T converging to ok we had seen that there is a one part if there is a whatever co-final family substituted restricted that it converges. Now in terms of subnets you have get if and only if let us look at if part that means what if you have a subnet converging then X must be a cluster point of S ok. So, let T equal to S composite P from E to X that is suppose this is subnet of S and let us assume that it converges straight take a neighborhood U of X there will exist some E 1 inside E such that E 1 is less than or equal to prime this is a relation inside E right E 1 less than or equal to E prime implies T E prime is inside U this is the convergence of T. Now given a D inside D I am going to now the original directed right first choose E inside E such that E prime less than or equal to E less than or equal to E prime implies D is less than or equal to P this is the property of P ok nothing more than that I am repeating that property here. Now take another element E 2 inside E such that this E 2 is following both E 1 and E ok that is possible because E is a directed set. So, E 1 is equal to E 2 and E is also less than or equal to E 2 it follows that D is less than or equal to P E 2 because because what P E E 2 is bigger than E. So, D is less than or equal to P E 2 and S composite P of E 2 which is T of E 2 that will be inside ok because as soon as something is something is bigger than E 2 T of that is inside U 2 is the first part here. Therefore, S is frequently inside U ok. Hence X is a cluster point of S ok. So, one way is done now the only part the only part you have to work little hard. Suppose X is a cluster point of S I have all to construct a subnet which converges to ok such a thing is not possible in an arbitrary topological space with sequences ok there may not be enough sequences at all right. So, so this is the this is something which the Nets have an advantage over sequences. So, let us see how it is done. We take ordered pairs D comma U belonging to D cross N X ok with the property that S of the first coordinate S of D is contained inside S of D belongs to you ok all right. What are you? You raise your neighborhood of it ok. So, obviously every element of N X is non-empty ok. So, this S has this property that S D is inside you. So, that is my definition of this set A. Now, what is the relation relation is again a strict relation coming from D cross N X. N X has a relation D has a relation take the strict relation namely D U less than or equal to prime D prime U prime if filled only if D is less than or equal to D prime that is the order of order of D you know direction in D and the direction in N X is by reverse inclusion U contains U prime ok. We need to check that this is a direction on E ok this is not true in general we have to use somehow the property of S namely X is a cluster point of S only because of that this is working ok. In any case transitivity reflexivity are obvious. So, this is there is no problem that does not depend upon S D they are true inside D cross N X. So, they will be true here also the problem is about the third point namely direction next that D 1 U on D 2 U 2 belong to A we must find out a D 3 U 3 right such that which is bigger than or follows both the two elements that is what we have to find out. So, first of all find a D 3 inside D such that D 1 and D 2 are less than or equal to D 3. So, this is because D is a directed. Now since X is a cluster point of S you will get a D 4 inside D such that D 4 less than or equal to D you would imply S D is inside U 1 intersection U 2 I am choosing U 1 intersection U 2 as a neighborhood of X for that I must have a D 4 D 4 less than or equal to D implies all such D's are inside all such S D's are inside U 1 intersection U 2 that just means that D 4 comma U 1 intersection U 2 is an element here because S D 4 itself is inside U 1 intersection U ok. So, I can put D equal D 4 here. So, that is there ok first of all this ordered pair is an element of it will be bigger than both D 1 U 1 and D 2 U 2 why because D I D 1 for example is less than or equal to D 4 D 1 is less than or equal to D 3 and D 3 is less than or equal to D 4 ok right. So, this is less than or equal to D 4 and U I are all bigger than U 1 intersection U 2 because U 1 and U 2 are there and I have taken an intersection this proves that this is a direction only ok. So, we have constructed a directed set we have yet to construct a net that will be a sub net of S ok for that I take P from E to D to be the first projection this is after all a product here take the first coordinate. So, it is an element of D. So, that is the function from E to D ok P is from E to D the first projection let us verify that P satisfies S 2 only then this P followed by S will be a I can take it S D that will be a sub net right. So, all that I have to do is P satisfies S 2 given a D inside D choose any x comma any D comma what some neighbor I have to choose I have chosen x D comma some neighbor I have to choose ok. So, choose the whole of x then this will be inside x already. So, that is an element of E then D comma x is less than or equal to D prime comma U just implies P of D prime is equal to D prime that D must be less than O D prime because D is less than O D prime is built in here ok. So, that is the meaning of S 2. So, once you have that I can take D equal to S composite P. So, we have constructed a sub net of S all that is fine, but we have to verify that this D converges to X starting with a cluster point of S we have got a sub net I want to show that it converges to X, but that is not difficult. Once again use the hypothesis that X is a cluster point of S given U inside an X there is a D inside D such that D prime first select D prime in D such that D is less than D prime and S D prime is inside U. So, that is because cluster point ok we have such a thing now D prime comma U inside E because S D prime is inside you this is a member of E now. If D prime U is less than or equal to A U prime take any member which is which follows D prime U then U prime will be a subset of U ok and T of A U prime by definition is S composite P of A U prime which is nothing but S of P A is A it is S A S A is inside U prime because once A is bigger than this D it will be inside U over therefore, T converges to X. So, thus we have proved what we have proved that criteria for a net to be having a cluster point a sub net must converge ok. So, that is the criteria it is easy to see that in a indiscreet space ok we have this property namely every sequence in it is convergent to every point right and you can characterize a indiscreet space by this property namely if every sequence converges to every point then it must be a indiscreet space. So, this is an easy thing which you must have seen in a part one it is also easy to see that in a discrete space on the contrary see I am taking these two extreme examples and what sequences can do to them ok. So, if you take discrete space every convergent sequence is eventually a constant right. However, the converse is false as seen in the earlier example we have seen that one right namely the uncountable set with the co-countable topology ok. Every convergent sequence in RCOC is eventually a constant yet RCOC is not discrete right. So, what is the role of a net here the net on the other hand has this property a space x is discrete if and only if every net which converges to a point x must assume the value x that is all but do not say eventually constant and so on just assuming the value if this happens for every net that space must be discrete. So, this is a powerful characterization of discrete spaces in terms of convergence of nets a space x is discrete if and only if every net which converges to a point must have the value x that is all the every sequence that is every net is not this one net that is all right. So, let us prove this one suppose x is discrete and s is a net converging to some point x ok. Then there must exist d belong into d such that d is less than to d prime implies s d prime is inside this neighborhood little x here singleton x is less than to it is a neighborhood of x because every singleton is open. So, I am choosing this neighborhood and for that I must get a get this property because of convergence d belong to d such that anything bigger than d d prime s d prime must be inside. In particular s of d also putting d prime equal to d here s of d must be x there is no other element here ok. So, the value is assumed over now converse now suppose x is not discrete then we will produce a sequence produce a net which converges and yet it does not assume that value at all. By the way there are many sequences which are convergent which do not assume the value which is the limit point right 1 by n for example. So, that is not a surprise you should realize that ok, but this ordinary character ordinary property characterizes the discrete space is something new out of net not out of sequences. So, you pay attention to that. So, let us prove the converse here suppose x is not discrete then there exist a point x such that singleton x is not open because if singleton x is open for every x then it is a discrete space. So, not discrete means there is one point at least such that singleton x is not open. In other words every neighborhood of x contains at least one other point. Therefore, we can construct a net now as from this neighborhood system n x to x which has the property that this s of u is inside u minus x. u minus x is not empty just by our observation namely our choice of x inside a non-discrete space right. So, this function again exists because right. This s converges to x because given any neighborhood u of x which is we can just take u itself and for everything which is which follows that u u is less than equal to v implies s of v must be inside v right because by choice here, but v is inside u. So, s to v will be inside u ok. So, this sequence does convert to x however we have seen that it never assumes the value x because it is always inside u minus x for some u for all the time ok. So, one can go on doing quite a bit of net theory, but I would like to stop here and then take up the study of filter and then as declared earlier we have seen we will study a little bit of net also in between namely whenever the properties of filter comes close to the properties of net also ok. In particular there will be many more results on filter than just what you see for net. Thank you.