In a branch of topology, namely the convexity spaces theory, the main objects that are investigated are infinite topological spaces where the closure operation $cl$ has the following property:
$$ cl(A)=\bigcup\limits_{F \subseteq A \\ F \, finite} cl(F) $$
Moreover, the following definition is classically provided: let $C$ be a closed set. Then a subset $B$ is said to span $C$ if $cl(B)=C$.
My questions are:
1) Are there examples of topological spaces such that each closed set is spanned by at least a finite set?
2) Moreover, are there examples of topological spaces such that each closed set is spanned only by finite sets?
1) $X$ equipped with indiscrete topology.
2) Every closed set is spanned by itself so if it is spanned only by finite sets then it must be finite. However the whole space $X$ is closed so if each closed set is spanned only by finite set then the whole space must be finite. In that case every closed subset of $X$ is finite hence is spanned by closed set (itself) and can only be spanned by finite sets (there are no sets available that are not finite).