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 $$\mathcal{T}:=\lbrace U\subseteq X\mid (x,y)\in C\mbox{ and }y\in U\mbox{ imply }x\to_U y\rbrace.$$ 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 $r\ge i$ and $s\ge j$ . Since $D$ is directed, there is a $k\in D$ such that $k\ge i$ and $k\ge j$ . It is clear that $x_k\in W$ and that any $t\ge k$ we have that $x_t\in W$ as well. Next, if $U_{\alpha}$ are sets in $\mathcal{T}$ , we want to show their union $U:=\bigcup \lbrace U_{\alpha}\rbrace$ 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 $F:=\prod \lbrace D_i \mid i\in D\rbrace$ , given by $z(i,f)=(i,f(i))$ .

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.