Monotone extenders for bounded c-valued functions
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 CA(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)→ CA(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. In this paper, vector spaces mean real vector spaces. Let X and Y be topological spaces. Then C(X,Y ) stands for the set of all continuous functions f : X → Y . If Y is a topological vector space, the set of all bounded continuous functions f : X → Y is denoted by C∞(X,Y ), where f : X → Y is bounded if f(X) is a bounded subset of Y , that is, for each 0-neighborhood U of Y there exists r ∈ R such that f(X) ⊂ rU . For A ⊂ X, a map u : C(A, Y ) → C(X,Y ) is called an extender if u(f)|A = f for each f ∈ C(A, Y ). For a topological vector space Y , an extender u : C(A, Y ) → C(X,Y ) is said to be a conv-extender (resp. conv-extender) if u(f)(X) is a subset of the convex hull (resp. the closed convex hull) of f(A) for each f ∈ C(A, Y ). For a topological space Y with a partial order structure ≤, an extender u : C(A, Y ) → C(X,Y ) (or u : C∞(A, Y ) → C(X,Y )) is said to be monotone if u(f) ≤ u(g) for each f, g ∈ C(A, Y ) (or f, g ∈ C∞(A, Y )) with f ≤ g. A vector space Y with a partial order structure ≤ is called an ordered vector space if the following axioms are satisfied: (O)1 x ≤ y implies x+ z ≤ y + z for all x, y, z ∈ Y ; (O)2 x ≤ y implies λx ≤ λy for all x, y ∈ Y and all λ > 0. A topological vector space Y that is also an ordered vector space is called an ordered topological vector space if the positive cone {y ∈ Y : y ≥ 0} is 2010 Mathematics Subject Classification: 46A40, 46A55, 46B40, 54C20, 54F05, 91A44.