Countable products of spaces of finite sets
We consider the compact spaces sigma_n(I) of subsets of an uncountable set I of cardinality at most n and their countable products. We give a complete classification of their Banach spaces of continuous functions and a partial topological classification.
A characterization of regular averaging operators and its consequences
We present a characterization of continuous surjections, between compact metric spaces, admitting a regular averaging operator. Among its consequences, concrete continuous surjections from the Cantor set C to [0, 1] admitting regular averaging operators are exhibited. Moreover we show that the set of this type of continuous surjections from C to [0, 1] is dense in the supremum norm in the set of all continuous surjections. The nonmetrizable case is also investigated. As a consequence, we obtain a new characterization of Eberlein compact sets.
Some open mapping theorems for measures
Given a compact Hausdorff space X, let C(X) be the Banach space of continuous real valued functions on X with sup norm and let M(X) be its dual considered as finite regular Borel measures on X. Let U(X) denote the closed unit ball of M(X) and let P(X) denote the nonnegative measures in M(X) of norm 1. A continuous map w of X onto another compact Hausdorff space Y induces a natural linear transformation 7T of M(X) onto M( Y) defined by setting 7([k)(g) = ,(g o q) for Z E M(X) and g E C(Y). It is shown that v is norm open on U(X) and on A P(X) for any subset A of the real numbers. If 9 is open, then X is weak* open on A P(X). Several examples are given which show that generalization in certain directions is not possible. The paper concludes with some remarks about continuous selections.