Completely Regular Space

A topological space $X$ 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 $f(A) = {0}$.

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

Further Reading

• Regular Spaces

See Also

• Normal Spaces
• Completely Normal Spaces
• Separation Axioms
• Countability Axioms
• Urysohn Lemma