Compactness is weakly hereditary
From Topospaces
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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Contents
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.
- 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: a compact space, a closed subset (given the subspace topology)
To prove: Consider an open cover of by open sets with , an indexing set. The 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 of such that , thus the union of the s contains . | is a subspace of | -- | -- | |
2 | The s along with form an open cover of | is closed in | -- | 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 s, that, along with , covers . | is compact | -- | Step (2) | |
4 | By throwing out , we get a finite collection of s whose union contains | -- | -- | Step (3) | |
5 | The corresponding now form a finite subcover of the original cover of . | -- | -- | 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)