Let $X$ be an arbitrary topological space, and $U,V\subseteq X$ two subspaces of $X$ such that $U\cong V$ ($U$ and $V$ are homeomorphic) with respect the subspace topology of $X$. I know examples where $U$ is closed in $X$ and $V$ is not. Is there some conditions to guarantee the next statement:
If $U$ is closed in $X$, then $V$ es closed in $X$.
If $X$ is compact and Hausdorff, then a subspace is closed if and only if it is compact. Since compactness is a topological invariant, $X$ being compact Hausdorff ensures that if two subspaces are homeomorphic and one is closed, then so is the other.