general topology - Limit sequence sets

In my measure theory book I came across the following definition:
Let $(A_n)_{n\ge1}$ be a sequence of subsets of some set $X$. Define:

$\limsup_{n\to\infty} A_n:=\bigcap_{n\ge1}\bigcup_{k\ge n}A_k$

$\liminf_{n\to\infty} A_n:=\bigcup_{n\ge1}\bigcap_{k\ge n}A_k$

Call the sequence convergent if $\limsup_{n\to\infty} A_n=\liminf_{n\to\infty} A_n$ , in which case we define $\lim_{n\to\infty} A_n:=\limsup_{n\to\infty} A_n$

My question is, does this notion of convergence correspond to some sort of metric on the set of subsets of $X$, or is it completely unrelated to the usual concept of a limit? Thanks