Selections and deleted symmetric products

Abstract

We give a very simple example of a connected second countable space $X$ whose hyperspace $[X]^{n+1}$ of unordered $(n + 1)$-tuples of points has a continuous selection, but $[X]^n$ has none. This settles an open question posed by Michael Hrušák and Ivan Martánez-Ruiz. The substantial part of the paper sheds some light on this phenomenon by showing that in the presence of connectedness this is essentially the only possible example of such spaces.