The Dugundji extension property can fail in \omega_\mu-metrizable spaces
by
Jerry E. Vaughan
University of North Carolina at Greensboro
Coauthors: Ian S. Stares

Let C(X) denote the set of all real valued continuous functions defined on a topological space X. A (necessarily normal) space X is said to have the Dugundji extension property (DEP) if for every closed subspace A of X there is a linear function P: C(A) --> C(X) (linear with respect to the usual vector space structure on C(A) and C(X)) such that P is an extender (i.e., P(f) extends the function f for all f in C(A)), and the range of P(f) is contained in the convex hull of the range of f for all f in C(A). J. Dugundji showed that every metric space has this property. A few generalized metric spaces also have this property (e.g., C. Borges showed that stratifiable spaces have the DEP). We show that there exist \omega_\mu -metrizable spaces which do not have the DEP (2^\omega₁ with the countable box topology is such a space). The proof given here is shorter than one previously announced. These examples answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false. In addition our technique also provides a simple proof of the known result that the Michael line does not have the DEP.

Date received: May 31, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caag-08.