Research Article

Banaschewski compactifications via special rings of functions in the absence of the Axiom of Choice

Published in: Quaestiones Mathematicae
Volume 48 , issue 7 , 2025 , pages: 1097–1125

DOI: 10.2989/16073606.2025.2467228

Author(s): Alireza Olfati Yasouj University, Iran , Eliza Wajch University of Siedlce, Poland

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

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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.