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Monotone extenders for bounded c-valued functions

100%
Yamazaki K.
Studia Mathematica
|
2010
|
tom 199
|
nr 1
17-22
EN
Let c be the Banach space consisting of all convergent sequences of reals with the sup-norm, $C_{∞}(A,c)$ the set of all bounded continuous functions f: A → c, and $C_{A}(X,c)$ the set of all functions f: X → c which are continuous at each point of A ⊂ X. We show that a Tikhonov subspace A of a topological space X is strong Choquet in X if there exists a monotone extender $u: C_{∞}(A,c) → C_{A}(X,c)$. This shows that the monotone extension property for bounded c-valued functions can fail in GO-spaces, which provides a negative answer to a question posed by I. Banakh, T. Banakh and K. Yamazaki.
2
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Extension of point-finite partitions of unity

63%
Ohta H., Yamazaki K.
Fundamenta Mathematicae
|
2006
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tom 191
|
nr 3
187-199
EN
A subspace A of a topological space X is said to be $P^{γ}$-embedded ($P^{γ}$(point-finite)-embedded) in X if every (point-finite) partition of unity α on A with |α| ≤ γ extends to a (point-finite) partition of unity on X. The main results are: (Theorem A) A subspace A of X is $P^{γ}$(point-finite)-embedded in X iff it is $P^{γ}$-embedded and every countable intersection B of cozero-sets in X with B ∩ A = ∅ can be separated from A by a cozero-set in X. (Theorem B) The product A × [0,1] is $P^{γ}$(point-finite)-embedded in X × [0,1] iff A × Y is $P^{γ}$(point-finite)-embedded in X × Y for every compact Hausdorff space Y with w(Y) ≤ γ iff A is $P^{γ}$-embedded in X and every subset B of X obtained from zero-sets by means of the Suslin operation, with B ∩ A = ∅, can be separated from A by a cozero-set in X. These characterizations are used to answer certain questions of Dydak. In particular, it is shown that, assuming CH, the property of A × [0,1] to be $P^{γ}$(point-finite)-embedded in X × [0,1] is stronger than that of A being $P^{γ}$(point-finite)-embedded in X.
3
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Extenders for vector-valued functions

51%
Banakh I., Banakh T., Yamazaki K.
Studia Mathematica
|
2009
|
tom 191
|
nr 2
123-150
EN
Given a subset A of a topological space X, a locally convex space Y, and a family ℂ of subsets of Y we study the problem of the existence of a linear ℂ-extender $u: C_{∞}(A,Y) → C_{∞}(X,Y)$, which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ ℂ, to bounded continuous functions f̅ = u(f): X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The results obtained allow us to characterize reflexive Banach spaces as the only normed spaces Y such that for every closed subset A of a GO-space X there is a ℂ-extender $u: C_{∞}(A,Y) → C_{∞}(X,Y)$ for the family ℂ of closed convex subsets of Y. Also we obtain a characterization of Polish spaces and of weakly sequentially complete Banach lattices in terms of extenders.
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