Dugundji Extenders on Ordered Spaces
by
Gary Gruenhage
Auburn University
Coauthors: Yasunao Hattori, Haruto Ohta

Let C(Y) denote the vector space of continuous real-valued functions defined on Y. If A subset X, a mapping \phi:C(A) --> C(X) is called a Dugundji extender if for every f in C(A)

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(i)" \phi(f) extends f;

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(ii)" \phi is linear;

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(iii)" the range of \phi(f) is contained in the convex hull of the range of f. J. Dugundji showed that such an extender exists whenever A is a closed subset of a metrizable space X. Versions of Dugundji's theorem for more general classes of spaces have been obtained by Borges, Heath and Lutzer, van Douwen, and others. In 1974, Heath and Lutzer asked if a Dugundji extender exists whenever A is a closed subset of a perfectly normal linearly ordered space. We give a negative answer to this question. We also discuss other extension properties for ordered spaces and show that in many cases (modulo measurable cardinals) the existence of a Dugundji-type extender is equivalent to A being a retract of a certain subspace of X.

Date received: February 13, 1998