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A characterization of almost continuity and weak continuity

Chrisostomos Petalas, Theodoros Vidalis (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

It is well known that a function f from a space X into a space Y is continuous if and only if, for every set K in X the image of the closure of K under f is a subset of the closure of the image of it. In this paper we characterize almost continuity and weak continuity by proving similar relations for the subsets K of X .

A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections

P. Holický, Miroslav Zelený (2000)

Fundamenta Mathematicae

Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then f - 1 ( y ) is a K σ set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of...

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