I'm a beginner in Topology. Today, this came up my mind:
(1) For a set $X$, choose a subset $A\subseteq X$. Let $S\subseteq X$ be a closed set if and only if $(A\subseteq S)\vee (S=\emptyset)$. This is a topology on $X$.
(2) For a set $X$, let $S\subseteq X$ be a closed set if and only if $(S $ is finite$)\vee (S = X)$. This is another topology on $X$.
The questions are:
 What are these two topologies called?
 Do they have significance? Are they used somewhere?
First question on stackexchange. Thanks!