TITLE: Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?
QUESTION [22 upvotes]: 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?
my ideas:
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 $X$
kind thanks

REPLY [4 votes]: Let $X'$ be a closed subset of $X$, and say that $H$ is closed in $X'$ in the subspace topology.  Then $X'\setminus H$ is open in $X'$, and is therefore of the form $G\cap X'$ for some set $G$ that is open in $X$.  Then $X\setminus G$ is closed in $X$, and $(X\setminus G)\cap X' = H$ is an intersection of sets closed in $X$ and is thus closed in $X$.