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Nets and Filters (Are Better Than Sequences)

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Nets and Filters (Are Better Than Sequences) Nets and filters (are better than sequences) Contents 1 Motivation 2 2 More implications we wish would reverse2 3 Nets 4 4 Subnets 6 5 Filters 9 6 The connection between nets and filters 12 7 The payoff 14 8 Filling in a gap from first year calculus 15 c 2018{ Ivan Khatchatourian 1 Nets and filters (are better than sequences) 2. More implications we wish would reverse 1 Motivation In the Section 5 of the lecture notes we saw the following two results: Theorem 1.1. Let (X; T ) be a Hausdorff space. Then every sequence in X converges to at most one point. Proposition 1.2. Let (X; T ) be a topological space, let A ⊆ X, and let a 2 X. If there is a sequence of points in A that converges to a, then a 2 A. We discussed how both of these implications really feel like they should reverse, but unfor- tunately neither of them do. In both cases, additionally assuming the topological space is first countable allows the implications to reverse. This is fine, but it still feels like sequences are not quite powerful enough to capture the ideas we want to capture. The ideal solution to this problem is to define a more general object than a sequence|called a net|and talk about net convergence. That is what we will do in this note. We will also define a type of object called a filter and show that filters also furnish us with a type of convergence which turns out to be equivalent to net convergence in all ways. With these more powerful tools in place of sequence convergence, both of the above implications will reverse, among many other benefits. We will not focus much on nets in this course (you will never be evaluated on them) but I still felt that I should mention them for the sake of completeness. We definitely will talk about filters later, so learning about them here will be helpful for you in the long run. We will also be able to fill in a big gap from first year calculus with this idea at the very end of this note. 2 More implications we wish would reverse In addition to the two results mentioned above that do not reverse, here is another one. It is essentially a restatement of the second result above in terms that are perhaps easier to digest. Proposition 2.1. Let (X; T ) be a topological space, let C ⊆ X be closed, and let x 2 X. If there is a sequence of points in C that converges to x, then x 2 C. Proof. Exercise. (This is essentially the same as the second result mentioned in the previous section.) This is another implication we wish would reverse. That is, it feels as though a set that contains all the limit points of all of its sequences should be closed. Again, this is unfortunately not the case (try to come up with a counterexample!). This property has a name though: Definition 2.2. Let (X; T ) be a topological space, and let A ⊆ X. A is said to be sequentially 1 closed if whenever a sequence fxngn=1 of elements of A converges to a point x, then x 2 A. c 2018{ Ivan Khatchatourian 2 Nets and filters (are better than sequences) 2. More implications we wish would reverse In this language, the proposition above says that every closed subset of a topological space is sequentially closed, though the converse is not true. Next, we give another example of an implication we wish would reverse. This requires another new definition beforehand. Definition 2.3. Let (X; T ) be an infinite topological space, and let A ⊆ X be infinite. A point x 2 X is called an accumulation point of A (more properly an !-accumulation point of A) if every open set containing x contains infinitely many points of A. So the property of being an accumulation point of A looks strictly stronger than the property of being in the closure of A. Instead of any open set around x simply intersecting A, we require this intersection to be infinite. For our purposes, we will restrict our attention to accumulation points of sequences. To be clear, given a sequence fxng, an accumulation point of the sequence is a point x such that any open set containing x contains xn for infinitely many n (but not necessarily a tail of the sequence). This all seems fine, so where is the problem? What implication does not reverse here? 1 Proposition 2.4. Let (X; T ) be a topological space and let fxngn=1 be a sequence in X (not 1 necessarily convergent). Let fxnk gk=1 be a subsequence of our sequence that converges to a point x. Then x is an accumulation point of the original sequence. 1 Proof. Let fxnk gk=1 be a subsequence as in the proposition, converging to x. Let U be an open set containing x. Then by definition of sequence convergence there is an N 2 N such that for all k ≥ N, xnk 2 U. Then the points xnN ; xnN+1 ; xnN+2 ; xnN+3 ;::: are all in U, and so x is an accumulation point of the sequence. This proposition is another implication we wish would reverse. It feels as though every accumulation point should be the limit of a subsequence. Again though, this implication fails. (Try to come up with a counterexample!) And again, if you additionally assume the space is first countable, the implication does reverse. By now I hope you have the feeling that first countability is a property designed to charac- terize the fact that sequence convergence determines what is going on in a topology. Getting a little more abstract, we have seen examples of very different topologies in which all the same sequences converge (though we have not yet remarked on it). Example 2.5. Consider the two spaces (R; Tco-countable) and (R; Tdiscrete). Then a sequence of real numbers converges in one of these spaces if and only if it converges in the other. Show this! So even though these are two very different-feeling topologies on the real numbers, it is impossible to detect any difference between them just be analyzing convergent sequences. This is good evidence that sequence convergence is not a powerful enough concept to encapsulate all the information a topology can carry, and that first countability is a property designed to patch this hole. The idea with net and filter convergence is to design a more powerful notion of convergence that obviates the need for such a patch. c 2018{ Ivan Khatchatourian 3 Nets and filters (are better than sequences) 3. Nets 3 Nets Recall that a sequence in a set X is simply a function x : N ! X. In order to generalize this idea, we are going to allow ourselves to consider functions whose domains are a more general sort of object, called a directed set. Definition 3.1. Let D be a set, and let ≤ be a relation on D satisfying the following properties: 1. ≤ is reflexive: for any x 2 D, x ≤ x. 2. ≤ is transitive: for any a; b; c 2 D, if a ≤ b and b ≤ c, then a ≤ c. 3. ≤ is directed: for any a; b 2 D there exists an element c 2 D such that a ≤ c and b ≤ c. A pair (D; ≤) satisfying these three properties is called a directed set. Example 3.2. Some examples of simple directed sets: 1. D = N with its usual ordering relation ≤. 2. D = f fn; n + 1; n + 2;::: g ⊆ N : n 2 N g, with the subset relation ⊆ or the superset relation ⊇. The latter is more useful, as we will soon see. 3. Let (X; T ) be a topological space, and let x 2 X. Then the set Dx := f U 2 T : x 2 U g is a directed set when equipped with the either the subset relation ⊆, or more usefully the superset relation ⊇. 4. If (D1; ≤1) and (D2; ≤2) are directed sets, then (D1 × D2; ≤) is a directed set where ≤ is defined by (a; b) ≤ (x; y) if and only if a ≤1 x and b ≤2 y: (Show this!) Definition 3.3. A net in a set X is a map w : D! X, where D is a directed set. Remark 3.4. Note immediately that a sequence is a net, since (N; ≤) is a directed set. Definition 3.5. If (X; T ) is a topological space and w : D! X is a net, we say that w converges to a point x 2 X if for any open set U containing x, there is a d 2 D such that Td := f w(e): d ≤ e 2 D g ⊆ U. We call a set of the form Td a tail of the net. If a net w converges to x, we denote this simply by w ! x, and refer to x a limit point of the net. c 2018{ Ivan Khatchatourian 4 Nets and filters (are better than sequences) 3. Nets Take a moment to verify to yourself that the use of the word \tail" in this context agrees with its use in the context of sequences, and that in the case where D = N with its usual ordering, this agrees with the usual definition of sequence convergence. Just to give some more language to use, we usually say that a net w : D! X is \eventually in A" if A contains some tail of w.
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