Dugundji extenders and retracts on generalized ordered spaces

Gary Gruenhage; Yasunao Hattori; Haruto Ohta

Fundamenta Mathematicae (1998)

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

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 c h -extender (resp. L c c h -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 c h -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 c h -extender φ:C(A) → C(X); and (vii) there is an L c c h -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].

How to cite

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Gruenhage, Gary, Hattori, Yasunao, and Ohta, Haruto. "Dugundji extenders and retracts on generalized ordered spaces." Fundamenta Mathematicae 158.2 (1998): 147-164. <http://eudml.org/doc/212308>.

@article{Gruenhage1998,
abstract = {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 $φ:C*_\{k\}(A) → C_\{p\}(X)$; (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].},
author = {Gruenhage, Gary, Hattori, Yasunao, Ohta, Haruto},
journal = {Fundamenta Mathematicae},
keywords = {Dugundji extension property; linear extender; π-embedding; retract; measurable cardinal; generalized ordered space; perfectly normal; product; GO-space},
language = {eng},
number = {2},
pages = {147-164},
title = {Dugundji extenders and retracts on generalized ordered spaces},
url = {http://eudml.org/doc/212308},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Gruenhage, Gary
AU - Hattori, Yasunao
AU - Ohta, Haruto
TI - Dugundji extenders and retracts on generalized ordered spaces
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 2
SP - 147
EP - 164
AB - 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 $φ:C*_{k}(A) → C_{p}(X)$; (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].
LA - eng
KW - Dugundji extension property; linear extender; π-embedding; retract; measurable cardinal; generalized ordered space; perfectly normal; product; GO-space
UR - http://eudml.org/doc/212308
ER -

References

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  14. [14] J. van Mill (ed.), Eric K. van Douwen Collected Papers, North-Holland, Amsterdam, 1994. Zbl0812.54001
  15. [15] K. Morita, On the dimension of the product of topological spaces, Tsukuba J. Math. 1 (1977), 1-6. Zbl0403.54021
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