Quotient Spaces Determined by Algebras of Continuous Functions

TL;DR

This paper proves that for certain locally compact spaces, the quotient topology determined by bounded continuous functions coincides with the completely regular topology, linking topological properties to algebraic structures.

Contribution

It establishes conditions under which the quotient topology and algebraically defined topology are equal for locally compact spaces, extending previous work on primitive ideal spaces.

Findings

Quotient topology equals the completely regular topology for certain spaces.

Second countability of the space implies second countability of the quotient.

Results connect topological properties with algebraic structures in $C^*$-algebras.

Abstract

we prove that if $X$ is a locally compact $\sigma$-compact space then on its quotient, $\gamma(X)$ say, determined by the algebra of all real valued bounded continuous functions on $X$, the quotient topology and the completely regular topology defined by this algebra are equal. It follows from this that if $X$ is second countable locally compact then $\gamma(X)$ is second countable locally compact Hausdorff if and only if it is first countable. The interest in these results originated in papers of R. J. Archbold, and S. Echterhoff and D. P. Williams where the primitive ideal space of a $C^*$-algebra was considered.