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
$$\mathcal{T}:=\{U\subseteq X\mid (x,y)\in C\text{and}y\in U\text{imply}x{\to}_{U}y\}.$$ Then $\mathcal{T}$ is a topology on $X$.
Proof.
Clearly $x{\to}_{X}y$ for any pair $(x,y)\in C$. In addition, $x{\to}_{\mathrm{\varnothing}}y$ 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 $U:=\bigcup \{{U}_{\alpha}\}$ 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 $D=\mathrm{dom}(x)$, and for each $i\in D$, we have that $({z}_{i},{x}_{i})\in \mathcal{C}$, with ${D}_{i}=\mathrm{dom}({z}_{i})$. 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}& \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & & & & & \hfill \mathrm{\ddots}\hfill & & & & & & & & & \\ \hfill \mathrm{\cdots}\hfill & \hfill {z}_{ia}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill {z}_{jf}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill {z}_{kp}\hfill & \hfill \mathrm{\cdots}\hfill & & & & & \hfill {z}_{if(i)}\hfill & & & & & & & & \\ \hfill \mathrm{\cdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\cdots}\hfill & & & & & & \hfill \mathrm{\ddots}\hfill & & & & & & & \\ \hfill \mathrm{\cdots}\hfill & \hfill {z}_{ib}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill {z}_{jg}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill {z}_{kq}\hfill & \hfill \mathrm{\cdots}\hfill & & & & & & & \hfill {z}_{jf(j)}\hfill & & & & & & \\ \hfill \mathrm{\cdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\cdots}\hfill & & & \hfill \Rightarrow \hfill & & & & & \hfill \mathrm{\ddots}\hfill & & & & & \\ \hfill \mathrm{\cdots}\hfill & \hfill {z}_{ic}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill {z}_{jh}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill {z}_{kr}\hfill & \hfill \mathrm{\cdots}\hfill & & & & & & & & & \hfill {z}_{kf(k)}\hfill & & & & \\ \hfill \mathrm{\cdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\cdots}\hfill & & & & & & & & & & \hfill \mathrm{\ddots}\hfill & & & \\ \hfill \mathrm{\cdots}\hfill & \hfill \downarrow \hfill & \hfill \mathrm{\vdots}\hfill & \hfill \downarrow \hfill & \hfill \mathrm{\vdots}\hfill & \hfill \downarrow \hfill & \hfill \mathrm{\cdots}\hfill & & & & & & & & & & & \hfill \searrow \hfill & & \\ \hfill \mathrm{\cdots}\hfill & \hfill {x}_{i}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {x}_{j}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {x}_{k}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \to \hfill & \hfill y\hfill & & & & & & & & & & \hfill y,\hfill & \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 onetoone 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  20130322 17:14:27 
Last modified on  20130322 17:14:27 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54A20 