Coherence of Smyth powerspaces

In this paper, we study when the Smyth powerspace $\mathcal{Q}^*_v(X)$ of a topological space $X$ is coherent, and prove that $X$ is coherent and weakly Hausdorff if and only if $\mathcal{Q}^*_v(X)$ is coherent and weakly Hausdorff.

We give examples to show that neither coherence nor weak Hausdorffness of $X$ solely implies that $\mathcal{Q}^*_v(X)$ is coherent or weakly Hausdorff.

As a byproduct, our work gives an affirmative answer to a question raised by Xu.