Tietze,s Extension Theorem
المؤلف:
Cullen, H. F.
المصدر:
Introduction to General Topology. Boston, MA: Heath, 1968.
الجزء والصفحة:
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29-7-2021
2299
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 
is normal iff, for all closed subsets 
of 
, every continuous function 
, where 
denotes the real line with the Euclidean topology, can be extended to a continuous function 
(Willard 1970, p. 103).
With respect to the alternative definition (Cullen 1968, p. 118), the statement is different: if 
is a T4-space, for all closed subsets 
of 
, every continuous bounded function 
can be extended to a continuous bounded function 
. (Cullen 1968, p. 127)
Another characterization of normality in terms of maps is Urysohn's lemma.
REFERENCES:
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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