α-Specker spaces

This paper is motivated by work on Specker spaces and a recent article of the authors on the ring of α-quotients. Here α denotes an uncountable regular cardinal or else ∞, indicating no cardinal constraint whatsoever. All spaces are compact Hausdorff, and for the most part zero-dimensional. strains of the “α-Specker” condition are studied: strong, weak, and one in between which is not qualified. One of the main results characterizes these conditions, for each α and each space X, in terms of the containment of C(X) in the ring of α-quotients of S(KαX), where the latter denotes the algebra of continuous functions with finite range, defined on an appropriate cover KαX of X. y c+-Specker” is equivalent to “Specker”. The paper examines the ω1-Specker conditions, proving that “weakly ω1-Specker” and “ω1-Specker” are equivalent. In fact, it is shown that X is weakly ω1-Specker if and only if for each f∈C(X) there is a Baire set B with meagre complement such that f(B) is countable.