Dugundji extenders and retracts on generalized ordered spaces

• Autorzy: Gary Gruenhage , Yasunao Hattori , Haruto Ohta
• Języki publikacji: EN

Abstrakty

EN

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\A; (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 $φ:C*_{k}(A) → C_{p}(X)$; (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 [9], van Douwen [1] and Hattori [8].

Słowa kluczowe

EN

Dugundji extension property; linear extender; π-embedding; retract; measurable cardinal; generalized ordered space; perfectly normal; product

Kategorie tematyczne

• 54B10: Product spaces
• 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
• 46E10: Topological linear spaces of continuous, differentiable or analytic functions
• 54C20: Extension of maps

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1998
• Tom: 158
• Numer: 2
• Strony: 147-164

Daty

wydano

1998

otrzymano

1997-07-02

poprawiono

1998-05-04

Twórcy

autor

Gary Gruenhage

• Department of Mathematics, Auburn University, Auburn, Alabama 36849, U.S.A.

autor

Yasunao Hattori

• Department of Mathematics and Computer Science, Shimane University, Matsue, Shimane, 690 Japan

autor

Haruto Ohta

Bibliografia

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