I have a non homework related question from a text and require a nice clear proof/disproof please
Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?
if $H$ is the subset of the topological space $X$
if the subset is closed in the closed subspace, the complement is open in the subspace, which means the complement is of form $U\cap H$ for some $U$ open in $X$
if the subspace is closed the complement is open which means complement of $H=U$ for some open $U$ in $X$