  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.
