Local compactness is weakly hereditary

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This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
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Property "Page" (as page type) with input value "{{{property}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
Property "Page" (as page type) with input value "{{{metaproperty}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

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

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.