A simultaneous selection theorem

Abstract

We prove a theorem that generalizes in a way both Michael's Selection Theorem and Dugundji's Simultaneous Extension Theorem. We use it to prove that if K is an uncountable compact metric space and X a Banach space, then C(K, X) is isomorphic to C(C, X) where C denotes the Cantor set. For X = R, this gives the well known Milyutin Theorem.

On a Theorem of Arvanitakis

“…Arvanitakis [2] established recently the following result extending both Michael's convex selection theorem [11] and Dugundji's simultaneous extension theorem [7]: Theorem 1.1. [2] Let X be a space with property c, Y a complete metric space and Φ : X → 2 Y a lower semi-continuous set-valued map with non-empty values.…”

“…Arvanitakis [2] established recently the following result extending both Michael's convex selection theorem [11] and Dugundji's simultaneous extension theorem [7]: Theorem 1.1. [2] Let X be a space with property c, Y a complete metric space and Φ : X → 2 Y a lower semi-continuous set-valued map with non-empty values. Then for every locally convex complete linear space E there exists a linear operator S : C(Y, E) → C(X, E) such that (1) S(f )(x) ∈ convf (Φ(x)) for all x ∈ X and f ∈ C(Y, E).…”

“…Arvanitakis [2] established recently the following result extending both Michael's convex selection theorem [11] and Dugundji's simultaneous extension theorem [7]:…”

“…Recall that a set-valued map Φ : X → 2 Y is lower semi-continuous if the set {x ∈ X : Φ(x) ∩ U = ∅} is open in X for any open U ⊂ Y . A space X is said to have property c [2] if X is paracompact and, for any space Y and a map φ : X → Y , φ is continuous if and only if it is continuous on every compact subspace of X. It is easily seen that the last condition is equivalent to X being a k-space (i.e., the topology of X is determined by its compact subsets, see [8]).…”

On a simultaneous selection theorem

“…for short) if for every open subset V of Y , the set {x ∈ X : Φ(x) ∩ V = ∅} is open in X. As a common generalization of Michael's convex-valued selection theorem [9] and Dugundji's simultaneous extension theorem [5], Arvanitakis [1,Theorem 1.1] established the following simultaneous selection theorem.…”