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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?

Emily
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  • (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) – Emily Feb 15 '23 at 04:51
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    Is $X$ assumed to carry a topology? I've never actually heard this called Day convolution by any semigroup theorist. – Benjamin Steinberg Feb 15 '23 at 14:20
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    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. – Benjamin Steinberg Feb 15 '23 at 14:23
  • @BenjaminSteinberg Apologies for the name, Day convolution is definitely non-standard terminology; I've been using it for quite a while and ended up forgetting that it was non-standard when I wrote the question (which I've rewritten now). – Emily Feb 15 '23 at 17:46
  • And thanks for the reference for Carruth's work! I couldn't find a statement for all of $\mathcal{P}(X)$, but Theorem 3.1 of Carruth's second book with Hildebrant has a proof that the subspace $C(X)$ of closed subsets of a compact semigroup $X$ is also a compact semigroup (I'm not sure if their proof applies if one drops compactness and replaces $C(X)$ with $\mathcal{P}(X)$, though). (They also mention Carruth's master thesis and Hofmann–Mostert's Elements of compact semigroups as sources that deal with this question, but I unfortunately don't have access to them) – Emily Feb 15 '23 at 17:46
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    (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) – Emily Feb 15 '23 at 17:47
  • The real expert is @TarasBanakh so maybe he will chime in. – Benjamin Steinberg Feb 15 '23 at 17:59
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    @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. – user95282 Feb 21 '23 at 14:18
  • @user95282 Thanks! I do find this very helpful =) When I originally asked this question I was mainly interested in better understanding the various notions of "continuous" relations/multivalued functions, though now I'm just trying to understand powerset topologies and how behaved they are in general. Essentially I'm looking for an interesting topology in $\mathcal{P}(X)$ making the usual operations on powersets continuous and which behaves well with the topology on $X$, which I've asked about here – Emily Feb 21 '23 at 18:58

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