## Camillo Costantini Weak orderability of some spaces which admit a weak selection

Comment.Math.Univ.Carolin. 47,4 (2006) 609-615.

Abstract:We show that if a Hausdorff topological space $X$ satisfies one of the following properties: \par \noindent a) $X$ has a countable, discrete dense subset and $X^2$ is hereditarily collectionwise Hausdorff; \par \noindent b) $X$ has a discrete dense subset and admits a countable base; \par \noindent then the existence of a (continuous) weak selection on $X$ implies weak orderability. As a special case of either item a) or b), we obtain the result for every separable metrizable space with a discrete dense subset.

Keywords: weak (continuous) selection, weak orderability, Vietoris topology, dense countable subset, isolated point, countable base, collectionwise Hausdorff space
AMS Subject Classification: Primary 54C65, 54F05; Secondary 54D15, 54D70, 54E35