- Tytuł:
- Dugundji extenders and retracts on generalized ordered spaces
- Autorzy:
-
Gruenhage, Gary
Hattori, Yasunao
Ohta, Haruto - Powiązania:
- https://bibliotekanauki.pl/articles/1205292.pdf
- Data publikacji:
- 1998
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
Dugundji extension property
linear extender
π-embedding
retract
measurable cardinal
generalized ordered space
perfectly normal
product - Opis:
- 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].
- Źródło:
-
Fundamenta Mathematicae; 1998, 158, 2; 147-164
0016-2736 - Pojawia się w:
- Fundamenta Mathematicae
- Dostawca treści:
- Biblioteka Nauki
Artykuł