I've encountered the sentences "topologized by convergence in probability" or "topologized by uniform convergences" and I would like to get an idea of the big picture.
Given notion of convergence on some set $X$ for which no topology has yet been specified (hence one cannot define convergence in the usual way), how it's possible to define a topology on $X$?
In this answer is said that it's possible to axiomatize the notion of convergence even without nets (and I'd be happy to not involve nets). Suppose one has a similar notion, it's enough to declare the closed sets as the sequentially closed to get a topology?
Best Answer
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.
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 $X$ can be described by sequences (which is true of first countable or metrizable spaces), then the above holds with nets replaced by sequences.
If $X$ is metrizable, then the above holds with net and subnet replaced by sequence and subsequence.