Urysohn's Lemma

A characterization of normal spaces which states that a topological space X is normal iff, for any two nonempty closed disjoint subsets A, and B of X, there is a continuous map f:X->[0,1] such that f(A)={0} and f(B)={1}. A function f with this property is called a Urysohn function.

This formulation refers to the definition of normal space given by Kelley (1955, p. 112) or Willard (1970, p. 99). In the statement for an alternative definition (e.g., Cullen 1968, p. 118), the word "normal" has to be replaced by T_4.

See also

Tietze's Extension Theorem, Urysohn's Metrization Theorem

This entry contributed by Margherita Barile

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Cullen, H. F. Introduction to General Topology. Boston, MA: Heath, p. 124, 1968.Joshi, K. D. "The Urysohn Characterization of Normality." §7.3 in Introduction to General Topology. New Delhi, India: Wiley, pp. 177-182, 1983.Kelley, J. L. General Topology. New York: Van Nostrand Company, p. 115, 1955.Willard, S. General Topology. Reading, MA: Addison-Wesley, p. 102, 1970.

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Urysohn's Lemma

Cite this as:

Barile, Margherita. "Urysohn's Lemma." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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