Abstract
We prove that if X is a locally compact σ-compact space, then on its quotient, γ(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 γ(X) is second countable locally compact Hausdorff if and only if it is first countable. The interest in these results originated in [1] and [7] where the primitive ideal space of a C*-algebra was considered.
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with an appendix by D. W. B. Somerset
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Lazar, A.J. Quotient spaces determined by algebras of continuous functions. Isr. J. Math. 179, 145–155 (2010). https://doi.org/10.1007/s11856-010-0075-0
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DOI: https://doi.org/10.1007/s11856-010-0075-0