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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?

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  • $\begingroup$ (P.S. Vietoris topologies are often defined only for subspaces of closed sets, but they also make sense for all of $\mathcal{P}(X)$; see e.g. section 1 of Clementino–Tholen) $\endgroup$
    – Emily
    Commented Feb 15, 2023 at 4:51
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    $\begingroup$ Is $X$ assumed to carry a topology? I've never actually heard this called Day convolution by any semigroup theorist. $\endgroup$
    – Benjamin Steinberg
    Commented Feb 15, 2023 at 14:20
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    $\begingroup$ My memory is that Carruth looked at this sort of thing and maybe you should look at his books with Hildebrant. The theory of topological semigroups. I think they are called hyperspace semigroups. $\endgroup$
    – Benjamin Steinberg
    Commented Feb 15, 2023 at 14:23
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    $\begingroup$ (I've also removed the second question; the idea was to find a smallest/largest assignment of topologies $\{\text{topologies on $X$}\}\to\{\text{topologies on $\mathcal{P}(X)$}\}$ for each set $X$ such that $x\mapsto\{x\}$ is an embedding, $f$ continuous implies $f_*$ continuous, and $\circledast$ is continuous, but trying to make this precise ran into set-theoretic issues) $\endgroup$
    – Emily
    Commented Feb 15, 2023 at 17:47
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    $\begingroup$ @Emily When X carries not just a topology but a uniform structure, a usually better behaved structure on P(X) is the hyperspace uniform structure. See e.g. "Uniform Spaces" by Isbell. This may or may not help, depending on your motivation for the question. $\endgroup$
    – user95282
    Commented Feb 21, 2023 at 14:18

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