Geodesic
On a weak form of uniform convergence
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 637-643

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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.
Classification : 40A30, 54E35, 54H05
Keywords: continuous functions on metric spaces; pointwise convergence; $\Delta $-convergence; analytic spaces; Hurewicz theorem; $K_\sigma $-spaces 
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Fuka, Jaroslav; Holický, Petr. On a weak form of uniform convergence. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 637-643. http://geodesic.mathdoc.fr/item/CMUC_2005__46_4_a3/