Compactness is not hereditary
This article gives the statement, and possibly proof, of a topological space property (i.e., compact space) not satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces).
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The property of topological spaces of being a compact space is not a subspace-hereditary property of topological spaces.
A subset of a compact space, equipped with the [[subspace topology], need not be a compact space.
- Compactness is weakly hereditary: Any closed subset of a compact space is compact.
- Hausdorff implies KC: Any compact subset of a Hausdorff space is closed. In particular, in a compact Hausdorff space, the compact subsets are the same as the closed subsets.
Consider the unit interval . This is a closed and bounded subset of the real line, hence it is compact. The subset , equipped with the subspace topology, is not compact, because it has an open cover given by subsets of the form that has no finite subcover. (Alternatively, it is not compact because it is not a closed subset of the real line).