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Completely Regular Space

A topological space X such that for every closed subset C of X and every point x in X\C, there is a continuous function f:X->[0,1] such that f(x)=0 and f(C)={1}.

This is the definition given by most authors (Kelley 1955, p. 117; Willard 1970, pp. 94-95). However, some authors (e.g., Cullen 1968, p. 130) require the additional condition that X be a T1-space. In any case, every completely regular space is regular, and the converse is not true.

See also

Completely Regular Graph, Tychonoff Space

This entry contributed by Margherita Barile

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References

Cullen, H. F. "Normal Spaces. Completely Regular Spaces." §18 in Introduction to General Topology. Boston, MA: Heath, pp. 118-140, 1968.Joshi, K. D. Introduction to General Topology. New Delhi, India: Wiley, p. 163, 1983.Kelley, J. L. General Topology. New York: Van Nostrand, 1955.Willard, S. "Regularity and complete regularity." §14 in General Topology. Reading, MA: Addison-Wesley, pp. 92-99, 1970.

Referenced on Wolfram|Alpha

Completely Regular Space

Cite this as:

Barile, Margherita. "Completely Regular Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CompletelyRegularSpace.html

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