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:

Step no.	Assertion/construction	Given data used	Facts used	Previous steps used	Explanation
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 $A$.	$A$ is a subspace of $X$	--	--
2	The $V_{i}$s along with $X\setminus A$ form an open cover of $X$	$A$ is closed in $X$	--	Step (1)	[SHOW MORE] $X\setminus A$ is open because $A$ is closed. Further, since the union of all the $V_{i}$s contains $A$, that along with $X\setminus A$ covers all of $X$.
3	The open cover from step (2) has a finite subcover. In other words, there is a finite subcollection of the $V_{i}$s, that, along with $X\setminus A$, covers $X$.	$X$ is compact	--	Step (2)
4	By throwing out $X\setminus A$, we get a finite collection of $V_{i}$s whose union contains $A$	--	--	Step (3)
5	The corresponding $U_{i}$ now form a finite subcover of the original cover of $A$.	--	--	Steps (1), (4)

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)