## Jaroslav Fuka, Petr Holick\'y

*On a weak form of uniform convergence *

Comment.Math.Univ.Carolinae 46,4 (2005) 637-643. **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

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