Compactness is weakly hereditary

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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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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

Weakly hereditary for properties related to compactness


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.


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_is contains A. A is a subspace of X -- --
2 The V_is along with X \setminus A form an open cover of X A is closed in X -- Step (1) [SHOW MORE]
3 The open cover from step (2) has a finite subcover. In other words, there is a finite subcollection of the V_is, 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_is 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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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)