Countable sets, BCO spaces and selections (Q1767713)

A base \({\mathfrak B}\) for a topological space \(X\) is called a base of countable order (BCO) if for every strictly decreasing sequence \(B_n\) of elements of \({\mathfrak B}\) containing \(x\), \(\{B_n: n= 1,2,\dots\}\) is a local base at \(x\). This paper is devoted to selections and multiselections of multifunctions taking values in BCO spaces (instead of metrizable). It is also proved that for a separable space \(X\), for each first countable space \(Y\) each multifunction \(F: X\to Y\) has a continuous selection if and only if \(X\) is countable.