Local compactness is weakly hereditary

Property-theoretic statement
The property of topological spaces of being locally compact satisfies the metaproperty of topological spaces of being weakly hereditary.

Verbal statement
Any closed subset of a locally compact space is locally compact.

Facts used

 * Compactness is weakly hereditary: Any closed subspace of a compact space is compact.

Proof
Given: A locally compact space $$X$$, a closed subset $$C$$

To prove: $$C$$ is locally compact

Proof: We need to show that given any point $$x \in C$$, there exists an open subset containing $$x$$ contained in a closed compact subset of $$C$$.

Since $$X$$ is locally compact, there exists an open set $$V \ni x$$ and a closed compact subset $$K$$ of $$X$$ containing $$V$$.

By the definition of subspace topology, $$V \cap C$$ is an open subset of $$C$$. Call this $$U$$. Further, $$K \cap C$$ is a closed subset of $$C$$. Call this $$L$$. We then have $$x \in U \subset L$$, with $$U$$ open and $$L$$ closed.

We need to show that $$L$$ is compact. For this, observe that $$L = K \cap C$$, and $$C$$ is closed in $$X$$, so $$L \subset K$$ is closed as a subset of $$K$$. Since any closed subset of a compact space is compact, we conclude that $$L$$ is compact, completing the proof.