Semi-continuous multifunctions and bases of countable order

This paper is devoted to the problem of selections. An important result in this area is Michaelʹs theorem on double selection for lower semi-continuous closed valued multifunctions. Recently we obtained a generalization of this theorem to a subclass of the so-called generalized metric spaces, namely the class of weakly developable spaces. One of the aims of this paper is to give an extension of our result (hence of Michaelʹs result) to a more general class of spaces, namely the class of spaces with a base of countable order. To do this, we give some results on spaces with a base of countable order which extend those of Wicke and Worrell Jr. Some applications are given. In particular we obtain a criterion of metrizability.