A \(T_0\)-Compactification Of A Tychonoff Space Using The Rings Of Baire One Functions

In this article, we continue our study of Baire one functions on a topological space \(X\), denoted by \(B_1(X)\) and extend the well known M. H. Stones's theorem from \(C(X)\) to \(B_1(X)\). Introducing the structure space of \(B_1(X)\), it is observed that \(X\) may not be embedded inside thi...

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Bibliographic Details
Published in:arXiv.org 2022-01
Main Authors: A Deb Ray, Mondal, Atanu
Format: Article
Language:English
Subjects:
Continuity (mathematics)
Embedded structures
Online Access:Get full text
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Summary:In this article, we continue our study of Baire one functions on a topological space \(X\), denoted by \(B_1(X)\) and extend the well known M. H. Stones's theorem from \(C(X)\) to \(B_1(X)\). Introducing the structure space of \(B_1(X)\), it is observed that \(X\) may not be embedded inside this structure space. This observation inspired us to build a space \(\mathcal{M}(B_1(X))/\sim\), from the structure space of \(B_1(X)\) and to show that \(X\) is densely embedded in \(\mathcal{M}(B_1(X))/\sim\). It is further established that it is a \(T_0\)-compactification of \(X\). Such compactification of \(X\) possesses the extension property for continuous functions, though it lacks Hausdorffness in general. Therefore, it is natural to search for condition(s) under which it becomes Hausdorff. In the last section, a set of necessary and sufficient conditions for such compactification to become a Stone-Ceck compatification, is finally arrived at.
ISSN:2331-8422