IPM - Institute for Research in Fundamental Sciences

“School of Mathematics”

Paper IPM / M / 18133

School of Mathematics
Title:	Banaschewski Compactifications via Special Rings of Functions in the Absence of the Axiom of Choice
Author(s):	Alireza Olfati (Joint with E. Wajch)
Status:	To Appear
Journal:	Quaestiones Math.
Supported by:	IPM

Abstract:

For a topological space

$\mathbf{X}$ ,

$U_{\aleph_0}(\mathbf{X})$ is the ring of all continuous real functions

$f$ on

$\mathbf{X}$ such that, for every real number

$\varepsilon>0$ , there exists a countable clopen cover

$\mathcal{A}$ of

$\mathbf{X}$ such that the oscillation of

$f$ on each member of

$\mathcal{A}$ is less than

$\varepsilon$ . For a zero-dimensional

$T_1$ -space

$\mathbf{X}$ , the ring

$U_{\aleph_0}(\mathbf{X})$ and its subring

$U_{\aleph_0}^{\ast}(\mathbf{X})$ of bounded functions from

$U_{\aleph_0}(\mathbf{X})$ are applied to necessary and sufficient conditions on

$\mathbf{X}$ to admit the Banaschewski compactification in the absence of the Axiom of Choice. For a zero-dimensional

$T_1$ -space

$\mathbf{X}$ and a Tychonoff space

$\mathbf{Y}$ , the problem of when the ring

$U_{\aleph_0}^{\ast}(\mathbf{X})$ can be isomorphic to

$U_{\aleph_0}^{\ast}(\mathbf{Y})$ or to the ring of all (bounded) continuous real functions on

$\mathbf{Y}$ is investigated. Several new equivalences of the Boolean Prime Ideal Theorem are deduced. Some results about

$U_{\aleph_0}(\mathbf{X})$ are obtained under the Principle of Countable Multiple Choices.

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