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., subspace-hereditary property of topological spaces)
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This article gives the statement, and possibly proof, of a basic fact in topology.

Statement

Any subset of a completely regular space is completely regular in the subspace topology.

Definitions used

Term	Definition used
completely regular space	A space $X$ is completely regular if it is a T1 space and given any point $x\in X$ and closed subset $A\subseteq X$ such that $x\notin A$, there exists a continuous map$f:X\to [0,1]$ such that $f(x)=0$ and $f(a)=1$ for all $a\in A$.
subspace topology	For a subset $A$ of the space $X$, the subspace topology on $A$ is defined as follows: a subset of $A$ is open in $A$ iff it can be expressed as the intersection with $A$ of an open subset of $X$. Also, a subset of $A$ is closed in $A$ iff it can be expressed as the intersection with $A$ of a closed subset of $X$.

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