Completely Regular Space

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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.


Normal spaces are completely regular
Completely regular spaces are regular

See Also


  1. Munkres, James R. Topology, 2015. pg. 209.