Countable sets, BCO spaces and 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.…”

“…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.…”

Continuous selections and σ-spaces

“…Then Φ has a continuous selection φ : X → Y . This result was extended in [17,Theorem 3.1], where it was proved that a space X is countable if, and only if, for each first-countable Y , each lsc multivalued map from X to Y has a continuous selection. In fact, their proof gives the following.…”

“…Theorem 2 (Yan and Jiang [17]). A separable space X is countable if and only if for each first-countable space Y and each open-valued lsc map Φ : X ⇒ Y , there is a continuous selection φ : X → Y .…”