Vol. 55, No. 2, 1974

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Dugundji extension theorems for linearly ordered spaces

Robert Winship Heath and David John Lutzer

Vol. 55 (1974), No. 2, 419–425

In this paper we prove a Dugundji Extension Theorem for a large class of monotonically normal spaces, the generalized ordered spaces. We show that if A is a closed subset of a generalized ordered space X and if C(A) and C(X) denote the vector spaces of continuous, bounded real-valued functions on A and X respectively, then there is a linear transformation u : C(A) C(X) such that for each g C(A), u(g) extends g and the range of u(g) is contained in the closed convex hull of the range of g. Furthermore, we give an example which shows that such linear transformations from C(A), the vector space of all continuous, real-valued functions on A, to C(X) cannot always be found, even when A is a closed, separable metrizable subspace of a hereditarily paracompact linearly ordered space.

Mathematical Subject Classification 2000
Primary: 54C20
Received: 28 March 1973
Published: 1 December 1974
Robert Winship Heath
David John Lutzer