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Hometopology via converging nets

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# 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 $x$

convergesto $y$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. 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

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

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.

## Mathematics Subject Classification

54A20*no label found*

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## Comments

## How to specify a "sequential" topology via its convergent se...

I found the article on "topology via net convergence" very interesting, but it also made me think of something: is it possible to do the same but restricting our attention only to sequences? For it happens often in Analysis that a certain topological structure is constructed by giving a definition of what it means for a sequence to converge (for example, in the definition of the topology in a space of distributions), and in such constructions it is rarely proven that the definition is in fact consistent with some actual topology, from which the convergence relation comes from...

Being more specific, my question is this: Â¿is there a set of conditions that are both necessary and sufficient for a relation pairing sequences with points in a space to be consistent with a topology in wich the relation is "convergence"?

## How to specify a "sequential" topology via its convergent se...

I found the article on "topology via net convergence" very interesting, but it also made me think of something: is it possible to do the same but restricting our attention only to sequences? For it happens often in Analysis that a certain topological structure is constructed by giving a definition of what it means for a sequence to converge (for example, in the definition of the topology in a space of distributions), and in such constructions it is rarely proven that the definition is in fact consistent with some actual topology, from which the convergence relation comes from...

Being more specific, my question is this: is there a set of conditions that are both necessary and sufficient for a relation pairing sequences with points in a space to be consistent with a topology in wich the relation is "convergence"?

## Re: How to specify a "sequential" topology via its convergen...

Yes, there is such a thing as a sequential convergence space

where the convergence structure is specified through convergent

sequences rather than open sets. As for the axioms defining

such a structure, I will post them later this evening because,

right now, I have to go somewhere.