Complete regularity is hereditary

This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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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