topology via converging nets

Given a topological space $X$ , one can define the concept of convergence of a sequence, and more generally, the convergence of a net. Conversely, given a set $X$ , a class of nets, and a suitable definition of “convergence” of a net, we can topologize $X$ . The procedure is done as follows:

Let $C$ be the class of all pairs of the form $(x,y)$ where $x$ is a net in $X$ and $y$ is an element of $X$ . For any subset $U$ of $X$ with $y\in U$ , we say that a net $x$ converges to $y$ with respect to $U$ if $x$ is eventually in $U$ . We denote this by $x\to_{U}y$ . Let

$\mathcal{T}:=\{U\subseteq X\mid(x,y)\in C\mbox{ and }y\in U\mbox{ imply }x\to_% {U}y\}.$

Then $\mathcal{T}$ is a topology on $X$ .

Proof.
Clearly $x\to_{X}y$ for any pair $(x,y)\in C$ . In addition, $x\to_{\varnothing}y$ is vacuously true. For any $U,V\in\mathcal{T}$ , we want to show that $W:=U\cap V\in\mathcal{T}$ . Since $x$ is eventually in $U$ and $V$ , there are $i,j\in D$ (where $D$ is the domain of $x$ ), such that $x_{r}\in U$ and $x_{s}\in V$ for all $r\geq i$ and $s\geq j$ . Since $D$ is directed, there is a $k\in D$ such that $k\geq i$ and $k\geq j$ . It is clear that $x_{k}\in W$ and that any $t\geq k$ we have that $x_{t}\in W$ as well. Next, if $U_{\alpha}$ are sets in $\mathcal{T}$ , we want to show their union $U:=\bigcup\{U_{\alpha}\}$ is also in $\mathcal{T}$ . If $y$ is a point in $U$ then $y$ is a point in some $U_{\alpha}$ . Since $(x,y)\in C$ with $x$ is eventually in $U_{\alpha}$ , we have that $x$ is eventually in $U$ as well. ∎

Remark. The above can be generalized. In fact, if the class of pairs $(x,y)$ satisfies some “axioms” that are commonly found as properties of convergence, then $X$ can be topologized. Specifically, let $X$ be a set and $C$ again be the class of all pairs $(x,y)$ as described above. A subclass $\mathcal{C}$ of $C$ is called a convergence class if the following conditions are satisfied

1.

$x$ is a constant net with value $y\in X$ , then $(x,y)\in\mathcal{C}$
2.

$(x,y)\in\mathcal{C}$ implies $(z,y)\in\mathcal{C}$ for any subnet $z$ of $x$
3.

if every subnet $z$ of a net $x$ has a subnet $t$ with $(t,y)\in\mathcal{C}$ , then $(x,y)\in\mathcal{C}$
4.

suppose $(x,y)\in\mathcal{C}$ with $D=\operatorname{dom}(x)$ , and for each $i\in D$ , we have that $(z_{i},x_{i})\in\mathcal{C}$ , with $D_{i}=\operatorname{dom}(z_{i})$ . Then $(z,x)\in\mathcal{C}$ , where $z$ is the net whose domain is $D\times F$ with $F:=\prod\{D_{i}\mid i\in D\}$ , given by $z(i,f)=(i,f(i))$ .

If $(x,y)\in\mathcal{C}$ , we write $x\to y$ or $\lim_{D}x=y$ . The last condition can then be visualized as

\begin{array}[]{cccccccccccccccccccc}&\vdots&\vdots&\vdots&\vdots&\vdots&&&&&% \ddots&&&&&&&&\\ \cdots&z_{ia}&\vdots&z_{jf}&\vdots&z_{kp}&\cdots&&&&&z_{if(i)}&&&&&&&\\ \cdots&\vdots&\vdots&\vdots&\vdots&\vdots&\cdots&&&&&&\ddots&&&&&&\\ \cdots&z_{ib}&\vdots&z_{jg}&\vdots&z_{kq}&\cdots&&&&&&&z_{jf(j)}&&&&&\\ \cdots&\vdots&\vdots&\vdots&\vdots&\vdots&\cdots&&&\Rightarrow&&&&&\ddots&&&&% \\ \cdots&z_{ic}&\vdots&z_{jh}&\vdots&z_{kr}&\cdots&&&&&&&&&z_{kf(k)}&&&\\ \cdots&\vdots&\vdots&\vdots&\vdots&\vdots&\cdots&&&&&&&&&&\ddots&&\\ \cdots&\downarrow&\vdots&\downarrow&\vdots&\downarrow&\cdots&&&&&&&&&&&% \searrow&\\ \cdots&x_{i}&\cdots&x_{j}&\cdots&x_{k}&\cdots&\to&y&&&&&&&&&&y,\end{array}

which is reminiscent of Cantor’s diagonal argument.

Now, for any subset $A$ of $X$ , we define $A^{c}$ to be the subset of $X$ consisting of all points $y\in X$ such that there is a net $x$ in $A$ with $x\to y$ . It can be shown that ${}^{c}$ is a closure operator, which induces a topology $\mathcal{T}_{\mathcal{C}}$ on $X$ . Furthermore, under this induced topology, the notion of converging nets (as defined by the topology) is exactly the same as the notion of convergence described by the convergence class $\mathcal{C}$ .

In addition, it may be shown that there is a one-to-one correspondence between the topologies and the convergence classes on the set $X$ . The correspondence is order reversing in the sense that if $\mathcal{C}_{1}\subseteq\mathcal{C}_{2}$ as convergent classes, then $\mathcal{T}_{\mathcal{C}_{2}}\subseteq\mathcal{T}_{\mathcal{C}_{1}}$ as topologies.

Title	topology via converging nets
Canonical name	TopologyViaConvergingNets
Date of creation	2013-03-22 17:14:27
Last modified on	2013-03-22 17:14:27
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 54A20