Urysohn's Lemma

A characterization of normal spaces which states that a topological space  is normal iff, for any two nonempty closed disjoint subsets , and  of , there is a continuous map  such that  and . A function  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 .

REFERENCES:

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.