Complete regularity is hereditary

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

Similar facts
Other similar facts:


 * Normality is weakly hereditary
 * Compactness is weakly hereditary

Opposite facts

 * Normality is not hereditary

Facts used

 * 1) uses::T1 is hereditary

Proof
Given: A topological space $$X$$, a subset $$A$$ of $$X$$. $$X$$ is completely regular.

To prove: $$A$$ is completely regular.

Proof: The T1 property for $$A$$ follows from Fact (1). It thus suffices to show the separation property for $$A$$, i.e., that any point and disjoint closed subset in $$A$$ can be separated by a continuous function. In other words, we want to prove the following.

To prove (specific): For any point $$x \in A$$ and any closed subset $$C$$ of $$A$$ such that $$x \notin A$$, there exists a continuous function from $$A$$ to $$[0,1]$$ such that $$f(x) = 0$$ and $$f(a) = 1$$ for all $$a \in C$$.

Proof (specific):

Textbook references

 * , Page 211-212, Theorem 33.2, Chapter 4, Section 33