Closed subset of closed subspace is closed

Statement

Suppose $X$ is a topological space and $B$ is a closed subset of $X$. Suppose $A$ is a subset of $B$ that is a closed subset of $B$ under the subspace topology on $B$ from $X$. Then, $A$ is also a closed subset of $X$.

Related facts

• Open subset of open subspace is open
• Dense subset of dense subspace is dense