Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects

Question

These are five important constructions and I would like to know how they are related.

The $n$th unordered configuration space of a space $X$ is $$ \operatorname{UConf}_n(X):=\{\text{embeddings of $\{1,...,n\}$ into $X$}\}/(\text{$n$th symmetric group}), $$ topologized as a subquotient of $X^n$.

The Ran space of $X$ is the set $\operatorname{Ran}(X)$ of finite subsets of $X$ with the topology generated by sets $$ \nabla_{U_1,...,U_n}:=\{S\in\operatorname{Ran}(U_1\cup\cdots\cup U_n)\mid S\cap U_i\ne\varnothing, i=1,...,n\} $$ where $U_i$ are disjoint open subsets of $X$.

The free topological semilattice $\operatorname{Sl}(X)$ on $X$ is the value on $X$ of the left adjoint to the forgetful functor from topological semilattices to topological spaces.

The Vietoris space $\mathscr VX$ of $X$ is the set of some (depending on the context) subsets of $X$ topologized by the same kind of $\nabla_{U_1,...,U_n}$ except that they are not required to be disjoint.

Finally, one may choose some nice embedding $I$ of some subcategory of spaces that contains $X$ into a topos in various ways, and consider there the power object $\Omega^{IX}$. Usually it is not in the image of $I$. There are versions like $\operatorname{Sub}_{\mathrm{fin}}(IX)\rightarrowtail\Omega^{IX}$ of objects of finite (say, Kuratowski finite) subobjects of $IX$ which might be. (Note that $\operatorname{Sub}_{\mathrm{fin}}$, with Kuratowski finiteness, is the free internal semilattice functor on any topos whatsoever.)

As a variation on the latter - say, $X$ is a simplicial set; since simplicial sets readily form a topos we have simplicial sets $\operatorname{Sub}_{\mathrm{fin}}(X)\rightarrowtail\Omega^X$.

Questions:

1. Is $\operatorname{UConf}_n(X)$ (homeomorphic to) a subspace of $\operatorname{Ran}(X)$?

2. There is a topology on $\bigcup_n\operatorname{UConf}_n(X)$ with $\{x_1,...,x_n,x_{n+1}\}$ close to $\{x_1,...,x_n\}$ when $x_{n+1}$ is close to $x_n$ in $X$. Is this homeomorphic to $\operatorname{Ran}(X)$?

3. The same two questions with $\operatorname{Sl}$ in place of $\operatorname{Ran}$.

4. Is $\operatorname{Ran}(X)$ homeomorphic to $\operatorname{Sl}(X)$?

5. Are $\operatorname{Ran}(X)$, $\operatorname{UConf}_n(X)$ or $\operatorname{Sl}(X)$ subspaces in $\mathscr VX$ for some nice spaces $X$?

6. Are there known embeddings of some categories of spaces into toposes such that the image of the embedding is closed under taking $\operatorname{Sub}_{\mathrm{fin}}$? In particular, can $\operatorname{Sub}_{\mathrm{fin}}(IX)$ be isomorphic to $I(\operatorname{Sl}(X))$ for some such $I$?

7. How does the geometric realization of $\operatorname{Sub}_{\mathrm{fin}}(X)$ relate to $\operatorname{Ran}$, $\operatorname{UConf}_n$, $\operatorname{Sl}$ and $\mathscr V$ of the geometric realization of $X$ for a simplicial set $X$?

$\ \ \,$0.$\ $Are these considered together and compared somewhere in the literature?

Answer 1

Too long for a comment but it is essentially a comment:

It is easy to see that for a Hausdorff space $X$ the topology on the Ran space coincides with the Vietoris topology and for a non-Hausdorff space $X$ the Ran topology is strictly weaker than the Vietoris topology.

The topology of the free topological semilattice is stronger than the Vietoris topology. For example, for an infinite compact metrizable space $K$ the space $SL(K)$ is a non-metrizable $k_\omega$-space whereas the Vietoris topology is metrizable. So, $SL(K)$ even topologically does not embed into $\mathcal V X$.

By the way, the hyperspace $\mathcal V X$ of non-empty finite subsets of a topological space $X$ endowed with the Vietoris topology coincides with the free Lawson topological semilattice of $X$.

For a Hausdorff space $X$ the configuration space $\mathrm{UConf}_n(X)$ naturally embeds into the free (Lawson) topological semilattice. For the free Lawson semilattice $\mathcal V X$ this can be shown by comparing the Vietoris (or Ran) topology with the quotient topology on $\mathrm{UConf}_n(X)$. Then combining this with the continuity of the natural maps $\mathrm{UConf}_n(X)\to SL(X)\to \mathcal V X$, we can conclude that $\mathrm{UConf}_n(X)\to SL(X)$ is an embedding, too.

Concerning the literature on the free (locally convex) topological semilattices (at least), you can look at the following papers and references therein:

Banakh and Sakai - Free topological semilattices homeomorphic to $\mathbb R^\infty$ or $\mathbb Q^\infty$

Banakh, Guran, and Gutik - Free topological inverse semigroups

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As I understand in question (2) on the space $\bigcup_{n\in\mathbb N}\mathrm{UConf}_n(X)$ it is considered the topology of direct limit of the tower $exp_n(X)$ where $exp_n(X)$ is the family of all at most $n$-element subsets of $X$ endowed with the Vietoris (or Ran) topology. This topology is stronger than the topology of $SL(X)$ and I am afraid that two topologies coincide only for $k_\omega$-spaces. For general spaces the operation of taking union is discontinuous with respect to this direct limit topology, so it is not a topological semilattice. This follows from Proposition 4, p. 35, of Banakh, Guran, and Gutik - Free topological inverse semigroups. This proposition says that if for a functionally Hausdorff space $X$ the free topological semilattice $SL(X)$ is a $k$-space, then each closed metrizable subspace of $X$ is locally compact.