Local compactness is weakly hereditary

This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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Statement

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.