Weak selections and weak orderability of function spaces
by
Valentin Gutev
University of KwaZulu-Natal, South Africa

Whenever X is a zero-dimensional space, the function space C_p(X, 2) has a continuous selection for its Vietoris hyperspace of at most 2-point sets (i.e., a continuous weak selection) if and only if X is separable. The result is applied to demonstrate that, for a strongly zero-dimensional metrizable space E, the function space C_p(X, E) is weakly orderable if and only if it has a continuous weak selection. This settles an open question of Tamariz-Mascarua, and provides a further partial positive answer to a question of van Mill and Wattel.

Date received: May 18, 2008