We know that A closed set in a metric space is $G_\delta$
Is there any topological space where a closed set is not necessarily $G_\delta$?
I am thinking a space where singletons are well known to be closed, but cannot be represented as countable intersection of open sets.
Example: The Sierpiński space $S=\{0,1\}$ with open sets $S,\ \emptyset,\ \{1\}.$ The set $\{0\}$ is closed but is not the intersection of any number of open sets.
Example: An uncountable set with the cofinite topology, i.e., the open sets are the cofinite sets and the empty set. Points are closed but are not $G_\delta$-sets. Of course this is not a Hausdorff space.
Example: The product of uncountably many compact Hausdorff spaces, each having at least two points. This is a compact Hausdorff space, and single points are not $G_\delta$-sets.