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:
Is $\operatorname{UConf}_n(X)$ (homeomorphic to) a subspace of $\operatorname{Ran}(X)$?
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 $\operatorname{Ran}(X)$?
Is $\operatorname{UConf}_n(X)$ (homeomorphic to) a subspace of $\operatorname{Ran}(X)$?
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 $\operatorname{Ran}(X)$?
The same two questions with $\operatorname{Sl}$ in place of $\operatorname{Ran}$.
Is $\operatorname{Ran}(X)$ homeomorphic to $\operatorname{Sl}(X)$?
Are $\operatorname{Ran}(X)$, $\operatorname{UConf}_n(X)$ or $\operatorname{Sl}(X)$ subspaces in $\mathscr VX$ for some nice spaces $X$?
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 $I$?
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 $X$?
The same two questions with $\operatorname{Sl}$ in place of $\operatorname{Ran}$$\ \ \,$0.
Is $\operatorname{Ran}(X)$ homeomorphic to $\operatorname{Sl}(X)$?
Are $\operatorname{Ran}(X)$, $\operatorname{UConf}_n(X)$ or $\operatorname{Sl}(X)$ subspaces in $\mathscr VX$ for some nice spaces $X$?
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 $I$?
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 $X$?
Are$\ $Are these considered together and compared somewhere in the literature?