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| Paper IPM / M / 18133 |
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| Abstract: | |
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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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