Existence of quasicontinuous selections for the space $2^{\Bbb R}$

Abstract: The paper presents new quasicontinuous selection theorem for continuous multifunctions $F X \longrightarrow\Bbb R$ with closed values, $X$ being an arbitrary topological space. It is known that for $2^{\Bbb R}$ with the Vietoris topology there is no continuous selection. The result presented here enables us to show that there exists a quasicontinuous and upper$\langle$lower$\rangle$-semicontinuous selection for this space. Moreover, one can construct a selection whose set of points of discontinuity is nowhere dense.

Keywords: continuous multifunction, selection, quasicontinuity

Classification (MSC91): 54C65, 54C08

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