Weak orderability of topological spaces

In 1951 Ernest Michael wrote a definitive seminal article on hyperspaces [E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951) 152–182] raising a general question that became known as Michaelʹs selection problem for hyperspaces. The present paper contains a detailed discussion on particular aspects of this problem, also some further open questions.