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Keywords:
weak (continuous) selection; weak orderability; Vietoris topology; dense countable subset; isolated point; countable base; collectionwise Hausdorff space
weak (continuous) selection; weak orderability; Vietoris topology; dense countable subset; isolated point; countable base; collectionwise Hausdorff space
Summary:
We show that if a Hausdorff topological space $X$ satisfies one of the following properties: \noindent a) $X$ has a countable, discrete dense subset and $X^2$ is hereditarily collectionwise Hausdorff; \noindent b) $X$ has a discrete dense subset and admits a countable base; \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.
References:
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