Every monoid $X$ induces a monoid structure $\circledast$ on $\mathcal{P}(X)$ via $$U\circledast V := \{uv\ |\ u\in U,v\in V\}.$$ Moreover, a morphism of monoids $f\colon X\to Y$ induces a morphism of monoids $f_*\colon\mathcal{P}(X)\to\mathcal{P}(Y)$ via the direct image.
Now, there are a couple of topologies we can put on $\mathcal{P}(X)$ making $\mathcal{P}(X)$ into a topological space and for which the continuity of $f$ implies the continuity of $f_*$, like the lower Vietoris, upper Vietoris, and Vietoris topologies on $\mathcal{P}(X)$.
Is it known whether the upper, lower, and Vietoris topologies also make $\mathcal{P}(X)$ into a topological monoid, i.e. such that $\circledast\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ is continuous?