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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Statement

Property-theoretic statement

The property of topological spaces of being a compact space is not a subspace-hereditary property of topological spaces.

Verbal statement

A subset of a compact space, equipped with the [[subspace topology], need not be a compact space.

Related facts

• 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.

Proof

Consider the unit interval $[0,1]$. This is a closed and bounded subset of the real line, hence it is compact. The subset $(0,1)$, equipped with the subspace topology, is not compact, because it has an open cover given by subsets of the form $(1/(n+2),1/n)$ that has no finite subcover. (Alternatively, it is not compact because it is not a closed subset of the real line).