Banaschewski compactifications via special rings of functions in the absence of the axiom of choice

Abstract

For a topological space X, U _ℵ0 (X) is the ring of all continuous real functions f on X such that, for every real number ε > 0, there exists a countable clopen cover A of X such that the oscillation of f on each member of A is less than ε. For a zero-dimensional T₁-space X, the ring Uℵ0 (X) and its subring U∗ℵ0 (X) of bounded functions from U_ℵ0 (X) are applied to necessary and sufficient conditions on X to admit the Banaschewski compactification in the absence of the Axiom of Choice. For a zero-dimensional T₁-space X and a Tychonoff space Y, the problem of when the ring U∗_ℵ0 (X) can be isomorphic to U∗_ℵ0 (Y) or to the ring of all (bounded) continuous real functions on Y is investigated. Several new equivalences of the Boolean Prime Ideal Theorem are deduced. Some results about U_ℵ0 (X) are obtained under the Principle of Countable Multiple Choices.

Mathematics Subject Classification (2020): Primary: 03E65, 54D35, 54C30, 54C40, 54D80;Secondary: 03E25, 54C20, 54C25, 54C35.