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We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.
The paper presents new quasicontinuous selection theorem for continuous multifunctions with closed values, being an arbitrary topological space. It is known that for with the Vietoris topology there is no continuous selection. The result presented here enables us to show that there exists a quasicontinuous and upperlower-semicontinuous selection for this space. Moreover, one can construct a selection whose set of points of discontinuity is nowhere dense.
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