# Dugundji extenders and retracts on generalized ordered spaces

Fundamenta Mathematicae (1998)

• Volume: 158, Issue: 2, page 147-164
• ISSN: 0016-2736

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## Abstract

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For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an ${L}_{ch}$-extender (resp. ${L}_{cch}$-extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous ${L}_{ch}$-extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and the pointwise convergence topology, for each space Y; (iv) A × Y is C*-embedded in X × Y for each space Y; (v) there is a continuous linear extender $\phi :C{*}_{k}\left(A\right)\to {C}_{p}\left(X\right)$; (vi) there is an ${L}_{ch}$-extender φ:C(A) → C(X); and (vii) there is an ${L}_{cch}$-extender φ:C(A) → C(X). We prove that these conditions are related as follows: (i)⇒(ii)⇔(iii)⇔(iv)⇔(v)⇒(vi)⇒(vii). If A is paracompact and the cellularity of A is nonmeasurable, then (ii)-(vii) are equivalent. If there is no connected subset of X which meets distinct convex components of A, then (ii) implies (i). We show that van Douwen’s example of a separable GO-space satisfies none of the above conditions, which answers questions of Heath-Lutzer , van Douwen  and Hattori .

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