Compactness is weakly hereditary

This article gives the statement, and possibly proof, of a topological space property (i.e., compact space) satisfying a topological space metaproperty (i.e., weakly hereditary property of topological spaces)
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Statement

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

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:

1. By the definition of subspace topology, we can find open sets $V_{i}$ of $X$ such that $V_{i}\cap A=U_{i}$, thus the union of the $V_{i}$s contains $A$.
2. Since $A$ is closed, we can throw in the open set $X\setminus A$, and get an open cover of the whole space $X$.
3. Since the whole space is compact, this open cover has a finite subcover. In other words, there is a finite subcollection of the $V_{i}$s, that, possibly along with $X\setminus A$, covers the whole of $X$.
4. By throwing out $X\setminus A$, we get a finite collection of $V_{i}$s whose union contains $A$. The corresponding $U_{i}$ now form a finite subcover of the original cover of $A$.

Proof in terms of finite intersection property

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References

Textbook references

• Topology (2nd edition) by James R. Munkres, ^{More info}, Page 165, Theorem 26.2, Chapter 3, Section 26
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe, ^{More info}, Page 12 (Theorem 4)