Working in an arbitrary topological space is it necessary that a closed set is the closure of some open set from this topology?
If not then what property does guarantee that any closed set is the closure of an open set?
Is a closed set the closure of an open set
general-topologyseparation-axioms
Best Answer
Just reposting my comment as an answer.
If $X$ is a $T_{1}$ space (a space in which all singletons are closed) that is not discrete, then $X$ has closed sets that are not the closure of an open set. In such a space, let $x$ be a point for which $\{x\}$ is not open. Then $\{x\}$ is a closed set by virtue of $X$ being $T_{1}$, but $\{x\}$ has empty interior and is therefore not the closure of an open set.
Being discrete is a sufficient property for every closed set to (trivially) be the closure of an open set, but I'm unsure of what a necessary and sufficient condition would be.