# Completely Regular Space

A topological space $(X,\tau)$ is said to be completely regular if one-point sets are closed in $X$ and if for each point $x_0$ and each closed set $A$ not containing $x_0$, there is a continuous function $f : X \to [0,1]$ such that $f(x_0) = 1$ and $f(A) = {0}$.^{[1]}

By the Urysohn Lemma, a normal space is completely regular, and a completely regular space is regular.

# Properties

Normal spaces are completely regular

Completely regular spaces are regular

# See Also

- Regular space
- Normal Space
- Completely Normal Space
- Separation Axioms
- Countability Axioms
- Urysohn Lemma

## References

- ↑ Munkres, James R. Topology, 2015. pg. 209.