Let $X$ be a countable set, and consider the topology of pointwise convergence of $\mathrm{Sym}(X)$, the symmetric group on $X$.
Then one can characterize the closed sets as follows: an element $g \in \mathrm{Sym}(X)$ is in a closed set $C$ if and only if for every finite subset $A \subset X$ there is a $g_A \in C$ for which $g$ and $g_A$ agree on $A$.
Is there a similar characterization for uncountable $X$?
It's the same characterisation for uncountable $X$. But the pleasant thing about the countable case is that $\text{Sym}(X)$ is completely metrisable, and for uncountable $X$ not first countable nor even normal. So much "uglier".