Continuous selections and σ-spaces

Abstract

MSC: primary 54C60, 54C65 secondary 26E25, 28B20 Keywords: σ -Space γ -Cover Fréchet filter Multivalued map Lower semicontinuity Clopen-valued map Continuous selection b-ScaleAssume that X ⊆ R \ Q, and each clopen-valued lower semicontinuous multivalued map Φ : X ⇒ Q has a continuous selection φ : X → Q. Our main result is that in this case, X is a σ -space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.

Extension of continuous functions and weakly developable spaces

“…Some results have been first obtained in [14] for Cechcomplete weakly k-developable spaces and generalized later in [3] for spaces having a monotonically complete base of countable order (BCO). New results on selection theory of set-valued mappings have been also obtained recently by several authors (see for example, [22,32,38]). As showed in [39], note that having a BCO in the selection theorem obtained in [38] for set-valued mappings defined on zero-dimensional paracompact spaces with values in spaces having a BCO can not be replaced by being stratifiable.…”