Closed Set in Topological Subspace/Corollary

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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 $T'$ if and only if $V$ 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 $T$.

$\blacksquare$


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