Abstract: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.
Keywords: continuous functions on metric spaces, pointwise convergence, $\Delta $-convergence, analytic spaces, Hurewicz theorem, $K_\sigma $-spaces
AMS Subject Classification: 54H05, 40A30, 54E35