Closed Set in Topological Subspace/Corollary

Theorem
Let $T$ be a topological space.

Let $T' \subseteq T$ be a subspace of $T$. Let subspace $T'$ be closed in $T$.

Then $V \subseteq T'$ is closed in $T'$ $V$ is closed in $T$.

Proof
Let $V \subseteq T'$ be closed in $T'$.

Then, from Closed Set in Topological Subspace, $V = T' \cap V$ is closed in $T'$.

If $V$ is closed in $T'$ then $V = T' \cap W$ where $W$ is closed in $T$.

Since $T'$ is closed in $T$, it follows by Topology Defined by Closed Sets that $V$ is closed in $T$.

Also see

 * Open Set in Open Subspace