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Baire classification of fragmented maps and approximation of separately continuous functions

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Abstract

We study a class of fragmented maps which contains all barely continuous maps, scatteredly continuous maps and maps which are pointwise discontinuous on each closed set. We prove that for any Hausdorff space X, a metric space Z and a fragmented map \(f:X\rightarrow Z\) there exists a pointwisely convergent to f sequence of continuous functions \(f_n:X\rightarrow Z\) with an additional condition (which is a weakening of the uniform convergence at each point) in the following cases: (a) X is stratifiable and Z is locally convex equiconnected; (b) X is stratifiable with \(\dim X<\infty \) and Z is equiconnected; (c) X is stratifiable with \(\dim X=0\). We show that for a hereditarily Baire space X, a compact space Y and a metric space Z every separately continuous map is a pointwise limit of a sequence of continuous maps which is uniformly convergent to f with respect to each variable at every point of in cases (a)–(c); (d) X is a perfect zero-dimensional compact space; and (e) X is the Sorgenfrey line.

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Acknowledgements

The authors thank the referees for their helpful and constructive comments that greatly contributed to improving the paper.

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  1. Department of Mathematical Analysis, Faculty of Mathematics and Informatics, Yurii Fedkovych Chernivtsi National University, Kotsyubyns’koho Str. 2, Chernivtsi, 58000, Ukraine

    Olena Karlova & Volodymyr Mykhaylyuk

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  1. Olena Karlova
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  2. Volodymyr Mykhaylyuk
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Correspondence to Olena Karlova.

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Karlova, O., Mykhaylyuk, V. Baire classification of fragmented maps and approximation of separately continuous functions. European Journal of Mathematics 3, 87–110 (2017). https://doi.org/10.1007/s40879-016-0123-3

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