Supplementary Exercises*: Nets

1. A directed set is a partially (strictly) ordered set $(J,\preceq)$ such that for any pair of elements $(\alpha,\beta)$ there is $\gamma$ such that $\alpha\preceq\gamma$ and $\beta\preceq\gamma$ .
2. We can require it to be a strict partial order or a weak partial order: the definitions and results below hold in either case
3. A cofinal subset $K$ of a partially ordered set $(J,\preceq)$ is a subset such that for any element in $J-K$ there is a greater element in $K$ .
4. If $J$ is a directed  set then its cofinal subsets are directed as well.
5. A net in a topological space $X$ is a function $f$ from a directed set $J$  into $X$ : $f=(x_{\alpha})_{\alpha\in J}$ where $x_{\alpha}=f(\alpha)$ .
6. In other words, a net is a subset of the space indexed by a directed set.
7. A sequence is a net.
8. A subnet of the net $(x_{\alpha})_{\alpha\in J}$ is $(x_{g(\beta)})_{\beta\in K}$ where $g:K\to J$ is such that $k\preceq k'\Rightarrow g(k)\preceq g(k')$ and $g(K)$ is cofinal in $J$ .
9. A subsequence is a subnet, but not every subnet of a sequence is a subsequence.
10. In fact, if a subnet was defined as a cofinal subset of a net then every subnet would be a subsequence, but then this definition would not be "flexible" enough for some criteria stated below. For example, a sequence may have a convergent subnet but no convergent subsequence.
11. The net $(x_{\alpha})$ converges to a point $x\in X$ if for each neighborhood $U$ of $x$ there is $\alpha$ such that $\alpha\preceq\beta$ implies $x_{\beta}\in U$ . We denote this by $x_{\alpha}\to x$ .
12. If $X$ is Hausdorff then a net cannot converge to more than one point.
13. If $x_{\alpha\in J}\to x\in X$ , $y_{\alpha\in J}\to y\in Y$ then $x_{\alpha}\times y_{\alpha}\to x\times y\in X\times Y$ .
14. A net $(x_{\alpha})$ converges to $x$ iff every its subnet converges to $x$ .
15. An accumulation point of net $(x_{\alpha})$ is a point $x$ such that for every its neighborhood $U$ the subset of indexes $\alpha$ such that $x_{\alpha}\in U$ is cofinal in $J$ .
16. A net has an accumulation point $x$ iff it has a subnet converging to $x$ .
17. The closure and nets.
18. $x\in\overline{A}$ iff there is a net of points of $A$ converging to $x$ .
19. Compare to the Sequence Lemma of Section 21:
20. if there is a sequence of points of $A$ converging to $x$ then the sequence is a net in $A$ and $x\in\overline{A}$
21. but if $x\in\overline{A}$ then there is just a net in $A$ converging to $x$ and this does not guarantee the existence of a sequence of points of $A$ converging to $x$ : something we can show if the space is metrizable
22. Continuous functions and nets.
23. $f:X\to Y$ is continuous iff for every convergent net $x_{\alpha}\to x$ : $f(x_{\alpha})\to f(x)$ .
24. Compare to Section 21:
25. if $f$ is continuous then the image of any net converging to any $x$ is a net (indexed by the same directed set) converging to $f(x)$ , in particular, this is true for any convergent sequence of points of $X$
26. but if the image of any sequence converging to any $x$ converges to $f(x)$ , this can still be not enough for the continuity of $f$ : it is enough if the space is metrizable
27. Compactness and nets.
28. $X$ is compact iff every net in $X$ has a convergent subnet.
29. Compare to Section 28:
30. if a space is compact then every sequence has a convergent subnet, but this does NOT imply that the sequence has a convergent subsequence
31. also if every sequence has a convergent subsequence this implies that every sequence has a convergent subnet but does not imply that every net has a convergent subnet
32. given these two, compactness and sequential compactness are not comparable
33. Let $X=\{0,1\}^{[0,1]}$ and $x_{n,j}$ be the nth digit of the binary expansion of $j$ (no tails of 1’s). $X$ is compact but not sequentially compact.
34. For any subsequence $\{n_{k}\}$ let $j$ be such that its $n_{k}$ ’s digit of the binary expansion is $k\mod2$ . Then $\{x_{n_{k},j}\}$ does not converge.
35. Let $B_{n}=\{x_{k}|k\ge n\}$ , $C_{n}=\overline{B_{n}}$ . Then $C=\cap_{n}C_{n}$ is not empty. Every point in $C$ is an accumulation point of the sequence. And vice versa. One cannot give an explicit example of such a point.
36. Topological groups (an application):
37. If $A$ is a closed subset and $B$ is a compact subset of a topological group $G$ then $A\cdot B$ is closed in $G$ .