# Article

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Keywords:
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.
References:
[1] Bartoszyński T., Judah H., Shelah S.: The Cichoń diagram. J. Symbolic Logic 58 (1993), 401-423. MR 1233917
[2] Fuka J.: On the $\delta$-convergence. Acta Universitatis Purkynianae 42, Czech-Polish Mathematical School, Ústí nad Labem, 1999, 63-64.
[3] Jech T.: Set Theory, Second Edition. Perspectives in Mathematical Logic, Springer, Berlin, 1997. MR 1492987
[4] Just W., Weese M.: Discovering Modern Set Theory. II. Graduate Studies in Mathematics, Vol. 18, American Mathematical Society, Providence, 1997. MR 1474727 | Zbl 0887.03036
[5] Kechris A.S.: Classical Descriptive Set Theory. Springer, New York, 1995. MR 1321597 | Zbl 0819.04002
[6] Martin D.A., Solovay R.M.: Internal Cohen extensions. Ann. Math. Logic 2 (1970), 143-178. MR 0270904 | Zbl 0222.02075
[7] Solovay R.M., Tennenbaum S.: Iterated Cohen extensions and Souslin's problem. Ann. of Math. 94 (1971), 201-245. MR 0294139 | Zbl 0244.02023
[8] Tichonov A.N.: On the regularization of ill-posed problems (Russian). Dokl. Akad. Nauk SSSR 153 (1963), 49-52. MR 0162378

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