Complete regularity is hereditary
This article gives the statement, and possibly proof, of a topological space property (i.e., completely regular space) satisfying a topological space metaproperty (i.e., hereditary property of topological spaces)
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This article gives the statement, and possibly proof, of a basic fact in topology.
Statement
Property-theoretic statement
The property of topological spaces of being completely regular is a hereditary property of topological spaces.
Verbal statement
Any subset of a completely regular space is completely regular in the subspace topology.
Definitions used
Completely regular space
Further information: completely regular space
Subspace topology
Further information: subspace topology
Proof
Proof outline
- Pick a point and a closed subset of the subspace
- Find a closed subset of the whole space, whose intersection with the subspace is the given subset
- Find a continuous function separating the point, and the bigger closed subset, in the whole space
- Restrict this continuous function to the subspace, and observe that this works
References
Textbook references
- Topology (2nd edition) by James R. Munkres, More info, Page 211-212, Theorem 33.2, Chapter 4, Section 33