Compactness is weakly hereditary

Property-theoretic statement
The property of topological spaces of being compact satisfies the metaproperty of being weakly hereditary: in other words, it is inherited by closed subsets.

Verbal statement
Any closed subset of a compact space is compact (when given the subspace topology).

Related facts

 * Hausdorff implies KC: In other words, every compact subset of a Hausdorff space is a closed subset.

Weakly hereditary for properties related to compactness

 * Paracompactness is weakly hereditary: Every closed subset of a paracompact space is paracompact.
 * Orthocompactness is weakly hereditary
 * Metacompactness is weakly hereditary

Proof in terms of open covers
Given: $$X$$ a compact space, $$A$$ a closed subset (given the subspace topology)

To prove: Consider an open cover of $$A$$ by open sets $$U_i$$ with $$i \in I$$, an indexing set. The $$U_i$$ have a finite subcover.

Proof:

Textbook references

 * , Page 165, Theorem 26.2, Chapter 3, Section 26
 * , Page 12 (Theorem 4)