Local compactness is weakly hereditary
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
- Compactness is weakly hereditary: Any closed subspace of a compact space is compact.
Given: A locally compact space , a closed subset
To prove: is locally compact
Proof: We need to show that given any point , there exists an open subset containing contained in a closed compact subset of .
Since is locally compact, there exists an open set and a closed compact subset of containing .
By the definition of subspace topology, is an open subset of . Call this . Further, is a closed subset of . Call this . We then have , with open and closed.
We need to show that is compact. For this, observe that , and is closed in , so is closed as a subset of . Since any closed subset of a compact space is compact, we conclude that is compact, completing the proof.