Abstract: We deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case -- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic order, and gives a characterisation of the so called weakly cyclically orderable spaces.
Keywords: Vietoris topology; continuous selection; weak selection; weakly orderable space; weakly cyclically orderable space
DOI: DOI 10.14712/1213-7243.2022.026
AMS Subject Classification: 54B20 54C65 54D05 54D30 54F05 54F65