 Welcome to NPTEL NOC, an introductory course on point set topology part 2. So, today we take up another topic in this chapter namely filters. So, today module 30, we will just study basics of filters. In a set X, it is easy to see that collection, the collection of all nets in X is too large and indeed not a set. On the other hand, filters are going to be families of power set PX of X and then the collection of all filters on X is a set. We shall see that filters will do quote unquote all the jobs that nets were invented for and some more through the concept of ultra filters and ultra close filters, these concepts or anything parallel to that is not available for nets. So, that is one way of looking at net versus filter. So, filter seems to have an advantage. So, that is one of the you may say justification for studying filters now after having studied nets quite thoroughly. Let X be a set by a filter or F, I will be using this F, F prime and so on this notation for filters. Filter F 1X, some people may say filter F in X also no problem. We mean a non-empty subset F of the power set PX satisfying the following three conditions. Empty set is not a member of F, please note this one, it is very important, F is closed under finite intersections, this one is familiar to you like in etymology, A contained inside X, these are subsets of X, A is in F, implies B is in F, all supersets of a member are also inside F, this is the condition. So, such condition was not there when you are studying topology at all. So, similar to conditions of topology, we have these three conditions, only these two, this is finite intersection is there and strongly empty set and X were there in the topology, this one denies that empty set is there, empty set should not be there. So, this is the important definition. Usually the name filter comes from the property 3, 1 and 2 seems to have been put there as an afterthought. This property 3 generalizes many, many areas and in many such areas also people use the word filters. So, there the definitions will be different in algebra, in geometry and various things you can use that. Note that 1 and 2 together imply that F has finite intersection property. I will be using this one again and again, what is the mean of finite intersection property given any set of finitely many members of F, their intersection should be non-empty. First of all 2 says their intersection is a member, whole gender finite intersection means take finitely many members here, take the intersection that also is a member, but that member is non-empty. So, together they imply F has finite intersection property. Since every filter is non-empty, you see started with a non-empty subset of F, in the definition of filter I have put a non-empty subset. As soon as a non-empty subset is there, X will be also there. So, I do not have to put that condition, X is there, empty set is not there and X is there. Also compare the conditions for a topology and for a filter, this is what I have done already. I will repeat it, a topology always has, always contains both empty set as well as whole set whereas property one says that empty set is not there. The second one is common to both of them, topology as well as filter. Third is a much stronger than the third property or whatever is namely arbitrary union of members of F is in, of tau is inside tau. So, that we have, looks like that we have replaced, but this is very strong. Even if one member is there, super sets, they will be union of those, they will be there. So, this is much more stronger than being a topology. We have seen that X is there, the total space, total set X is there for all filters F inside X. This implies that the smallest and simplest filter is singleton X. You just take singleton X, of course X should not be empty set, X itself should not be empty set. Then singleton X is a filter, because it has Y and Y identification property and there is nothing more bigger than that, so this is all. On the other hand, the collection of all non-empty subsets of X is not a filter unless X itself is a singleton. You see, in the case of topology, there was this singleton X as well as, sorry, X as well as empty set. If they are there, that was the indiscrete space. If we allow all the subsets, that is a discrete space. Here, we cannot allow all of them at all. Throw away the empty set, yet whatever is left out, that will not be a filter unless X itself is a singleton, because as soon as X has two points, you can take singleton X, singleton Y, they will be disjoint, but their intersection is empty, so that is not allowed. The finite intersection property will not be valid or property 2 here will not be valid. If two disjoint sets are there, so that is the important thing you have to observe here. So this is one example. Let us see some more useful examples. Given any non-empty subset X, subset of X, look at the family of A, this notation we will use again and again. Of all subsets of B, subsets B of X, which contain A, including A of course, this filter is called an atomic filter with A as its atom, one single element and all its one single subset and all supersets. So that is a filter obviously and that filter is called atomic filter with A as its atom. Now there is a notational thing here. If A is a singleton A, I will not put that bracket here. I will just write F little a for F capital A. Now suppose you have an infinite set. Then there is just like in the case of topology, there is this co-finite filter. What is it? Don't take empty set but take all other members of tau, this co-finite topology, namely all subsets such that their complement is finite, co-finite subsets. So that will be automatically a filter because first of all if you take any two members, their intersection has to be non-empty because both of them have complement. Complement is finite and you are working in an infinite set, that is important. So moment one is there, anything which is bigger than that will also co-finite, it will be complement will be finite. So that is also there. So this is an example not very useful or anything just to make the concepts of filter a little more clear what is happening. Given a net S from D to X, so D is a directed set, S is from D to X, take any element A inside D and take the section S A, set of all S B such that B follows A that or B is bigger than A whatever you want to say including A also. Look at that, that is the definition little S A here. Now look at this S is coming here, I am defining this F S, F S is all subset A of X such that there exist an A belong to D such that this S A is contained inside A. Including the equality is allowed here, all S S will be there and their supersets will be there. That will be very easy to verify that that itself is a filter. So this filter is called the filter associated to the net S. So I am bringing net and filter together here via this concept namely S going to F S. This filter is called the filter associated to a net S. It is important because it plays the role of a one way bridge from nets to filter. Why I am calling one way? I do not know any nice way of going from filters to the nets. This way it is very nice, very easy work. So this one way bridge is there. We shall have an opportunity to elaborate on this point. Why this important bridge? Just by having such a function does not make much sense. So you will see why this is important. Here is one more example in a topological space. Take the family of all neighbourhoods and X of a point X. Given each X you take the neighborhood of. That X is fixed here by that. This N X itself is a filter. Finite, intersections, all of them will contain X so it is non-empty. Anything bigger than that will be neighborhood so automatically they are all there and so on. The whole space is there. MTZ is not there because MTZ cannot be a neighborhood. So you may say that this simple example is the role model for the topological theory of filters. Remember the filters were defined without any reference to topology. There was no topology on X but what we did was we compared it with the concept of topology also that is all. So far there was no topology. If this example is the first example wherein topology of X is used to get a filter. And this filter is the filter which guides us as far as the topological theory. What do you mean by this one? Convergence and such thing. For this we have to refer to this one. Let us see how it comes. Now before taking up the study of interaction of filters with topologies, let us proceed our study of filters in a fashion similar to study of topologies but independent of any reference to particular topologies on X. Except noting down certain similarities between this and this similarities between this and the concept of topology that is all. So we will take more examples also later on. So first notice that if F alpha is a family of filters on X then their intersection is a filter. You had a similar theorem for topology. That is what I wanted to say that we are comparing. Intersection of arbitrary families of filters. All of them are remember they are sub-families of Px. So whole thing it makes sense. What is the meaning of this? Take all subsets A of X such that if A is things are F alpha for every alpha. So that is an intersection of this family. That is a filter. Of course empty set is not in any of them. So it will not be here also. So you can easily verify that it is a filter. However even the union of just two filters may fail to be a filter because one filter may have an element of some set another one. Their intersection may be empty. You do not know that. There is no hypothesis. That is one reason why union of two filters may fail to be a filter. This was there in topology also. But luckily if we have two topologies we can take the union and then generate a topology containing that. Even that will not be possible in the case of filters. So however we would like to do that kind of thing here namely generating filter. Just the way we have done generating topology. A sub-family B of a filter is called a base for F if for each A inside B there exist a B belonging to B such that B is contained inside A. This is very very similar to the definition of a base for a topology. There for each X and a member containing that there must be something B here. Here there is no reference to the point at all. For each A inside F there must be a smaller B which is inside this curly B. In other words take a member of B, curly B, take all supersets. They can be they are allowed to be inside F. That is the construction here now. That is where we are leading the world. So we have made the definition of a base here. For example the entire F is a base for itself just like in a topology. The entire F is a base. So this definition of a base satisfies similar to the conditions of what we have done in the what we have done for topology but it is much more general. For example the base for a topology had to have this property namely union of all members is X. It must be a cover for X. No such condition is here. Let us look at some examples. Any local base at X belongs to X where X is a topological space now is a base for the filter NX. So this was one of the important filter. You do not have to take all the neighbourhoods. You take a base in the usual terminology of a local base in a topology. That will be a base in the sense of filter. It is a base for the filter. Because what is it if you have a base for every neighbourhood of X where you remember in the base. That condition is satisfied. That is what it is here. A base determines a unique filter in the following way. The family of all sets which contain some member of B. You see there may be a filter here. B is a base for that. But this filter what I am generating may be even larger. Why because this family of all sets which contain some member of B is taken. In particular if I start a filter and a base for it the filter generated by that determined by that may be even larger. So it is clear that if we start with a non-empty family B which satisfies one and two. Then it defines a unique filter for which B is a base. Also it is clear that different bases may give rise to same filter. If you rise to means I have given you I have described a method here. Generating or whatever you want to call. Taking all the members which are supersets of some member of B. The following result gives a complete picture now. A non-empty B contained inside PX of non-empty subsets of X. It is a base for some filter on X. If we will down live B1, B2 belong to B. There exists B3 belonging to B such that B3 is contained inside B1 intersection B2. You see this is again similar to the base similar to topology. Except here no reference to the point. Given any two things intersection should contain another member. Intersection may not be there. Intersection should contain another member from B3. So once this is there because they are non-empty. So finite intersections will be also non-empty. And finite intersections you can go B1, B2, B3, B3, B1, B2. Then you keep taking further and further intersections of that. If B3, B4 is there then you can take B5 which is contained in that one and so on. So this is very easy to prove by induction. So non-empty family B of BX, non-empty subsets of X is a base for some filter. If we will down live for B1, B2 belong to B. There exists a B3 belonging to B. So this is similar to that one. There is no problem with that one. Now we come to the next stage of generating. And then only I will write use the word generating here. Exactly similar to what we did in topology. What is the next stage? Sub-base. What is a sub-base? Let us define. A sub-family S of BX is called a sub-base for a filter. If we will down live the family of all finite intersections of members of S forms a base. You understand? See there is no condition of finite intersections and so on that was taken care by this condition on the base. Now starting with arbitrary family S that will called a sub-base. This is a definition now for a filter F if and only if the family of all finite intersections form a base. In this case we say S generates F or equivalently F is generated by F. Okay? Yeah tell me. This condition that family of all finite intersections form a base. This automatically implies that every that empty set cannot mean the S. Empty set can be there. Empty set will not be there but the family may be empty. Empty set will not be a member of this one at all. Because if empty set is there then it will be there in the base also. Right? But it cannot be a base because once it is in the base it will be in the filter also. But that is not allowed. S also cannot be empty set. S can be empty. Why? Because what is the finite intersection of members of S? Members of an empty set. That will be complete X. All X's will be there. Once X is the only thing now. Okay? Singleton X forms a base for what? Singleton X. If S is empty then we will get our smallest filter. Smallest filter. The smallest filter has two bases, sub bases. Empty set is a sub base. Singleton X is also a base but it is a base. Singleton X is the whole space also. Okay? Singleton X is not a sub base. For sub base we need non-empty. Non-empty families. Okay? Because once it is sub base we are taking only supersets there. We are not going to take finite intersections there. So empty set is not a member of any base. Whereas empty set cannot be a member but the family itself can be empty. Okay? So as we have pointed out, finite intersection of members of S forms a base. Automatically implies that empty set is not a member. But this family S may have no members also. That is allowed. However, in practice it gives you only the singleton X, right? Therefore, for all practical purposes you can assume start with S as non-empty family. So family S contains a pf is a sub base for a filter if and only if S has finite intersection property. Once again you may be confused. If S is empty set it has finite intersection property. Because the only family which is finite family is empty family there. The empty family intersection is the whole set. That is not non-empty. Okay? So now we will do little more with functions. This is again similar to what we did in topology. Start with any function from one set to another set. S B S sub base for a filter F on X. Then I want to push it to Y. What is how do I push it? Take F of S curly S equal to all F A where A is inside S. So this is a sub family of P Y, right? If S has some elements there then F A will have some elements, alright? So this family is a sub base for a filter and that filter will be denoted by F check of F. Okay? So how do I do that? What is the meaning of sub base for a filter now? Take finite intersections of members of F S then take all the supersets. So that is your F check of F, alright? So all that I have to see is that this family has finite intersection property. If you take F A1, F A2, F An intersection is F of A1, A2, A1 intersection that will be contained inside that point. So that is non-empty. That is all. Notice that the family F F, okay? Namely if you take S instead of S here put F, the whole F where F is a filter, okay? F F that is F A is F A fails to be a filter on Y in general, okay? What we have done? You take a sub base and then this will be a sub base for a filter. It will not, it may not be a filter in general because there may be bigger subsets which do not come from S at all, right? Bigger subsets which do not come from even F. So if you take F A such that A is inside F, it may fail to be a filter, okay? However, if you take F check of F that contains the family of F because what is the meaning of F check of F? F check of F, treat F as a sub base for F itself, right? Then F check of F makes sense. So this is, this contains FF, that is all. This is similar to situation of push-outs of topology. Same thing we have done for topologies also. If you take open subsets of X and then take their images, okay? That may fail to become a topology. But unlike in the case of topology, pulling back filters going the other way, okay? Where it was nice in the case of topology? Pulling back filters under arbitrary function does not work here. Remember if Y is a topological space, that is a topology, X is some set and F from X to Y is a some function. Then I can take inverse image of open subsets in Y, they themselves form a topology on X. That was a nice situation for. Here it does not work, not only that, we do not need it at all, okay? So even it is worse than that. Namely even the family F inverse of B where B is a, B since S prime is a sub-base, okay? This may not satisfy the finite intersection property. So of course if we assume F is subjective, then this works. So you have to assume more and more hypothesis. So let us not bother about because we are not going to use it in the convergence theory of filters which is our AM after all, okay? So this is what I mean by saying that basics of filters, that is all. Next time we shall now study seriously its relation with topology, namely convergence theory for filters, okay? Thank you.