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Keywords:
continuous functions on metric spaces; pointwise convergence; $\Delta $-convergence; analytic spaces; Hurewicz theorem; $K_\sigma $-spaces
continuous functions on metric spaces; pointwise convergence; $\Delta $-convergence; analytic spaces; Hurewicz theorem; $K_\sigma $-spaces
Summary:
The notion of $\Delta $-convergence of a sequence of functions is stronger than pointwise convergence and weaker than uniform convergence. It is inspired by the investigation of ill-posed problems done by A.N. Tichonov. We answer a question posed by M. Kat\v{e}tov around 1970 by showing that the only analytic metric spaces $X$ for which pointwise convergence of a sequence of continuous real valued functions to a (continuous) limit function on $X$ implies $\Delta $-convergence are $\sigma$-compact spaces. We show that the assumption of analyticity cannot be omitted.
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