Continuous selections and sigma-spaces

TL;DR

This paper explores the relationship between continuous selections of multivalued maps and the topological property of being a sigma-space, providing new insights and reformulations related to the Scheepers Conjecture.

Contribution

It establishes that if every clopen-valued lower semicontinuous multivalued map from a metrizable separable space to Q has a continuous selection, then the space is a sigma-space, and offers a partial converse.

Findings

Spaces with continuous selections are sigma-spaces

Partial converse results are provided

Reformulation of Scheepers Conjecture in terms of continuous selections

Abstract

Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.