Topology induced by a convergence notion

First, you need a topological version of convergence. I could impose that a sequence of real numbers $(x_n)$ converge to $0$ if $x_1$ is odd. This is not a topological notion of convergence because the constant sequence $(0,0,0,0,\ldots)$ does not converge to $0$. (And we know that, in topology, a constant sequence must converge to the constant.) As Lucio points out, there are incredibly useful forms of convergence that are not topological, such as almost sure (or almost everywhere) convergence of a sequence of random variables (or measure functions).

We will describe a topological notion of convergence in terms of nets for the sake of generality. (One could also use filters.) If we describe convergence in terms of sequences, we are restricting how many topologies we could create because there are topologies in which sequential convergence is not sufficient.

The following is from Problem 11D in Willard's General topology textbook.

Suppose we have some notion of convergence on a set $X$ satisfying the following properties. Fix $x\in X$ and let $I$ be a directed set.

(a) If $x_i=x$ for each $i\in I$, then the net $(x_i)$ converges to $x$.

(b) If $(x_i)$ converges to $x$, then every subnet of $(x_i)$ converges to $x$.

(c) If every subnet of $(x_i)$ has a subnet converging to $x$, then $(x_i)$ converges to $x$.

(d) (Diagonal principle) If $(x_i)$ converges to $x$ and, for each $i\in I$, a net $(x^i_j)_{j\in J_i}$ converges to $x_i$, then there is a diagonal net converging to $x$; i.e., the net $(x^i_j)_{i\in I,\,j\in J_i}$, ordered lexicographically by $I$, then by $J_i$, has a subnet which converges to $x$.

Then if the closure of a subset $E$ of $X$ is defined by $$ \overline{E}:=\{x\in X \mid x_i\to x\ \text{for some net $(x_i)$ contained in $E$}\}, $$ the result is a topological space in which the notion of net convergence is as originally specified.

The moral of the story is that a topological notion of convergence tells you which sets are closed, and thus which sets are open. So we have a topology. In analysis, it is very common to describe and use topologies in this way. Since working with the open sets directly may be difficult, you may recall some useful theorems that rely only on convergence.

If $X$ and $Y$ are topological spaces and $f:X\to Y$ is a function, then $f$ is continuous iff whenever a net $(x_i)$ in $X$ converges to a point $x$ in $X$, then $f(x_i)\to f(x)$ in $Y$.

If $X$ can be described by sequences (which is true of first countable or metrizable spaces), then the above holds with nets replaced by sequences.

A topological space $X$ is compact iff every net in $X$ has a subnet which converges.

If $X$ is metrizable, then the above holds with net and subnet replaced by sequence and subsequence.