Research Article
Banaschewski compactifications via special rings of functions in the absence of the Axiom of Choice
DOI:
10.2989/16073606.2025.2467228
Keywords:
Primary: 03E65, 54D35, 54C30, 54C40, 54D80, Secondary: 03E25, 54C20, 54C25, 54C35, Banaschewski compactification, maximal Tychonoff compactification, ultra-filter, ring of continuous functions, maximal ideal, weak forms of the Axiom of Choice,
Abstract
For a topological space 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 of X such that the oscillation of f on each member of is less than ϵ. For a zero-dimensional T
1-space X, the ring and its subring of bounded functions from 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 T
1-space X and a Tychonoff space Y, the problem of when the ring can be isomorphic to 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 are obtained under the Principle of Countable Multiple Choices.
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