It can be proved that arbitrary union of open sets is open. Suppose $v$ is a family of open sets. Then $\bigcup_{G \in v}G = A$ is an open set.
Based on the above, I want to prove that an arbitrary intersection of closed sets is closed.
Attempted proof: by De Morgan's theorem:
$(\bigcup_{G \in v}G)^{c} = \bigcap_{G \in v}G^{c} = B$. $B$ is a closed set since it is the complement of open set $A$.
$G$ is an open set, so $G^{c}$ is a closed set.
$B$ is an infinite union intersection of closed sets $G^{c}$.
Hence infinite intersection of closed sets is closed.
Is my proof correct?
