Completely Regular Space
المؤلف:
Cullen, H. F.
المصدر:
Normal Spaces. Completely Regular Spaces." §18 in Introduction to General Topology. Boston, MA: Heath
الجزء والصفحة:
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16-7-2021
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Completely Regular Space
A topological space 
such that for every closed subset 
of 
and every point 
, there is a continuous function ![f:X->[0,1]](https://mathworld.wolfram.com/images/equations/CompletelyRegularSpace/Inline5.gif)
such that 
and ![f(C)=<span style=]()
{1}" src="https://mathworld.wolfram.com/images/equations/CompletelyRegularSpace/Inline7.gif" style="height:15px; width:64px" />.
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 
be a T1-space. In any case, every completely regular space is regular, and the converse is not true.
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.
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