A $T_0$-Compactification Of A Tychonoff Space Using The Rings Of Baire One Functions

TL;DR

This paper extends classical theorems to Baire one functions, constructing a $T_0$-compactification of a Tychonoff space that generalizes the Stone-Cech compactification, with conditions for Hausdorffness.

Contribution

It introduces a new $T_0$-compactification using Baire one functions and characterizes when it becomes a Stone-Cech compactification.

Findings

Constructed a $T_0$-compactification of $X$ from Baire one functions.

Proved $X$ is densely embedded in this compactification.

Identified conditions for the compactification to be Hausdorff and coincide with the Stone-Cech compactification.

Abstract

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.