Tietze's Extension Theorem

A characterization of normal spaces with respect to the definition given by Kelley (1955, p. 112) or Willard (1970, p. 99). It states that the topological space X is normal iff, for all closed subsets C of X, every continuous function f:C->R, where R denotes the real line with the Euclidean topology, can be extended to a continuous function F:X->R (Willard 1970, p. 103).

With respect to the alternative definition (Cullen 1968, p. 118), the statement is different: if X is a T4-space, for all closed subsets C of X, every continuous bounded function f:C->R can be extended to a continuous bounded function F:X->R. (Cullen 1968, p. 127)

Another characterization of normality in terms of maps is Urysohn's lemma.

This entry contributed by Margherita Barile

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Cullen, H. F. Introduction to General Topology. Boston, MA: Heath, 1968.Joshi, K. D. "The Tietze Characterization of Normality." §7.44 in Introduction to General Topology. New Delhi, India: Wiley, pp. 182-188, 1983.Kelley, J. L. General Topology. New York: Van Nostrand, 1955.Willard, S. General Topology. Reading, MA: Addison-Wesley, pp. 99-108, 1970.

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Tietze's Extension Theorem

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Barile, Margherita. "Tietze's Extension Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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