# 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. 1.

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

2. 2.

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

3. 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. 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 TopologyViaConvergingNets 2013-03-22 17:14:27 2013-03-22 17:14:27 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 54A20