Regular Space

According to most authors (e.g., Kelley 1955, p. 113; McCarty 1967, p. 144; Willard 1970, p. 92) a regular space is a topological space in which every neighborhood of a point contains a closed neighborhood of the same point.

Another equivalent condition is the following: for every closed set C and every point x not in C there are two disjoint open sets U and V such that C subset= U and x in V.

In other sources (e.g., Bourbaki 1989, p. 80; Cullen 1968, p. 113) regularity is defined differently, using separation axioms.

See also

Completely Regular Space

This entry contributed by Margherita Barile

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Bourbaki, N. "Regular Spaces." §1.4 in Elements of Mathematics: General Topology. Berlin: Springer-Verlag, pp. 80-81, 1989.Cullen, H. F. "Regular Spaces." §17 in Introduction to General Topology. Boston, MA: Heath, pp. 113-118, 1968.Kelley, J. L. General Topology. New York: Van Nostrand, 1955.McCarty, G. "Regularity and T_3-Spaces." In Topology, an Introduction. New York: McGraw-Hill, pp. 144-146, 1967.Willard, S. "Regularity and Complete Regularity." §14 in General Topology. Reading, MA: Addison-Wesley, pp. 92-99, 1970.

Referenced on Wolfram|Alpha

Regular Space

Cite this as:

Barile, Margherita. "Regular Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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