Coherence of Smyth powerspaces

Abstract

In this paper, we study when the Smyth powerspace ${\mathcal{Q}}_v^{\ast }(X)$ of a topological space $X$ is coherent, and prove that $X$ is coherent and weakly Hausdorff if and only if ${\mathcal{Q}}_v^{\ast }(X)$ is coherent and weakly Hausdorff. We give examples to show that neither coherence nor weak Hausdorffness of $X$ solely implies that ${\mathcal{Q}}_v^{\ast }(X)$ is coherent or weakly Hausdorff. As a byproduct, our work gives an affirmative answer to a question raised by Xu.