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NETS There Are a Number of Footnotes, Which Can Safely Be Ignored NETS There are a number of footnotes, which can safely be ignored, especially on the first reading. 1. Nets, convergence, continuity Nets are generalization of sequences needed to deal with convergence in general topological spaces, where convergent sequences are not enough to describe the topology. The idea is to replace the set of natural numbers, which is used to index elements of sequences, by arbitrary sets such that it makes sense to talk about \large" elements. Recall that a partial order on a set I is a binary relation ≺ such that (a) i ≺ i; (b) if i ≺ j and j ≺ i; then i = j; (c) if i ≺ j and j ≺ k; then i ≺ k: Definition 1.1. A set I with a partial order1 ≺ is called directed if for any i; j 2 I there exists k 2 I such that i ≺ k and j ≺ k. The following example will be our main source of directed sets. Example 1.2. Let X be a set and I be a collection of subsets of X closed under finite intersections, so if A; B 2 I then A \ B 2 I. Define a partial order on I by A ≺ B iff B ⊂ A: Then I is a directed set: for any A; B 2 I, we have A ≺ A \ B and B ≺ A \ B. Definition 1.3. A net in a set X is a collection (xi)i2I of elements of X indexed by a nonempty directed set I, or in other words, it is a map I ! X. Definition 1.4. We say that a net (xi)i2I in a topological space X converges to an element x 2 X, and write xi ! x or simply xi ! x, if for every neighbourhood U of x there exists i0 2 I such that xi 2 U for i 2 all i i0. More generally, we say that x is a cluster point of (xi)i2I if for every neighbourhood U of x and every i0 2 I there exists i i0 such that xi 2 U. In a Hausdorff topological space a net can converge to at most one point, which we then denote by lim xi. i Example 1.5. Assume X is a topological space, x 2 X, and I is the collection of all neighbourhoods of x. Since I is closed under finite intersections, we can order it by the inverse inclusion as in Example 1.2. For every U 2 I choose an element xU 2 U. Then (xU )U2I is a net in X converging to x, since for every neighbourhood U of x we have xV 2 V ⊂ U for all V U. Proposition 1.6. Assume X is a topological space, A ⊂ X and x 2 X. Then x 2 A¯ if and only if there exists a net in A converging (in X) to x. Proof. ) If x 2 A¯, then for every neighbourhood U of x there exists an element xU 2 U \A. By Example 1.5 this gives us a net (xU )U converging to x. ( If xi 2 A and xi ! x, then for every neighbourhood U of x there exists i0 such that xi 2 U for all ¯ i i0. In particular, A \ U 6= ;. Since this is true for all U, we conclude that x 2 A. Date: November 23, 2017. 1More often than not, it is assumed that ≺ is only a preorder, or quasiorder, that is, condition (b) is omitted. But I follow [M] here. 2The terms limit point and accumulation point are also used, but then they should not be confused with limit/accumulation points of the set fxi j i 2 Ig ⊂ X. 1 Corollary 1.7. Assume F1 and F2 are two topologies on a set X such that for any net (xi)i in X and any 3 x 2 X we have xi ! x in topology F1 if and only if xi ! x in topology F2. Then F1 = F2. Proof. By the previous proposition, for any set A ⊂ X and any x 2 X, the element x belongs to the closure of A in topology F1 if and only if it belongs to the closure of A in topology F2. It follows that a set is closed in topology F1 if and only if it is closed in topology F2. Hence the same is true for open sets, so F1 = F2. Proposition 1.8. A map f : X ! Y between topological spaces is continuous at a point x 2 X if and only if for every net (xi)i in X converging to x we have f(xi) ! f(x). Proof. ) Take a neighbourhood V of f(x). We have to find i0 2 I such that f(xi) 2 V for all i i0. By continuity of f at x, there exists a neighbourhood U of x such that f(U) ⊂ V . As xi ! x, there exists i0 2 I such that xi 2 U for all i i0. Then f(xi) 2 V for all i i0. ( Suppose that f is not continuous at x. This means that there exists a neighbourhood V of f(x) such that for every neighbourhood U of x we have f(U) 6⊂ V . Hence, for every such U, there exists an element −1 xU 2 U n f (V ). Then, again by Example 1.5, we get a net (xU )U converging to x, yet f(xU ) 62 V for all U, so the net (f(xU ))U does not converge to f(x). 2. Subnets Subnets are analogues of subsequences. An immediate idea is to say that a subnet of a net (xi)i2I is determined by a subset J ⊂ I with some nice properties. This turns out to be too naive. For example, it would be impossible to prove Proposition 2.2 below using such a notion. The correct definition is less straightforward. Definition 2.1. A net (yj)j2J is called a subnet of a net (xi)i2I if there exists a map ': J ! I such that 4 yj = x'(j) and for every i0 2 I there exists j0 2 J such that '(j) i0 for all j j0. Proposition 2.2. Assume x is a cluster point of a net (xi)i2I in a topological space X. Then there exists a subnet of (xi)i converging to x. Proof. Consider the set J of pairs (i; U), where i 2 I and U is a neighbourhood of x such that xi 2 U. Define a partial order on J by (i1;U1) ≺ (i2;U2) iff i1 ≺ i2 and U2 ⊂ U1: Let us check that J is directed. Take two points (i1;U1) and (i2;U2) in J. As I is directed, there exists 0 0 0 0 k 2 I such that i1 ≺ k and i2 ≺ k . Since x is a cluster point, there exists k k such that xk 2 U1 \ U2. Then (i1;U1) ≺ (k; U1 \ U2) and (i2;U2) ≺ (k; U1 \ U2): Next, define a map ': J ! I by '(i; U) = i. Let us check that ' has the property required in Def- inition 2.1. Take i0 2 I and a neighbourhood U0 of x. We can find k0 i0 such that xk0 2 U0. Then j0 = (k0;U0) is an element of J with the property that '(j) i0 for all j j0. 3A bit more informally we can say that a topology is completely determined by specifying convergence of nets. A natural question then is whether we can start with a rule of convergence instead of open sets to develop a meaningful theory. This is indeed possible and leads to the notion of convergence spaces. Convergence spaces are more general than topological spaces, so not every rule of convergence is defined by a topology, but they have found much fewer applications. 4This is only one of possible definitions, due to Kelley. The definition in [M], due to Willard, is different and less convenient to work with. More precisely, a Willard subnet is a Kelley subnet, but not every Kelley subnet is a Willard subnet. If you are bothered by the fact that there are different notions of subnets, the good news is that they have the same functionality in the following rigorous sense. Two nets (yj )j2J and (zk)k2K are said to be AA-equivalent (AA stands for Aarnes and Andenæs - two mathematicians from Trondheim), if for every j0 2 J there exists k0 2 K such that fzk j k k0g ⊂ fyj j j j0g and, conversely, for every k0 2 K there exists j0 2 J such that fyj j j j0g ⊂ fzk j k k0g. It is easy to see that AA-equivalent nets converge to the same elements and have the same cluster points. Now, it is possible to show that any Kelley subnet of a net (xi)i is AA-equivalent to a Willard subnet of (xi)i. 2 Thus, by letting yj = x'(j) we get a subnet (yj)j2J of (xi)i2I . It remains to show that yj ! x. Let U be a neighbourhood of x. There exists i0 2 I such that xi0 2 U. Consider j0 = (i0;U) 2 J. Then for any j = (i; V ) j0 we have yj = x'(j) = xi 2 V ⊂ U: Hence yj ! x. 3. Compactness Theorem 3.1. For any topological space X the following conditions are equivalent: (i) X is compact; (ii) every net in X has a cluster point; (iii) every net in X has a convergent subnet.
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