Manifolds of sets

Question

Consider a set $M$ such that $M$ is a set of all countable subsets of a compact set $\Omega \subset \mathbb R^d$.Let $d(x, y)$ be the distance between two sets $x, y \in M$. For instance this could be the Hausdorff distance. My question is: is the set $M$ a manifold? If yes, then how do I find a diffeomorphism between $M$ and a subset of $\mathbb R^m$ for some $m$? Or even show that such a diffeomorphism exists.

Answer 1

Take $d=1$ and $\Omega = [0,1]$. Fix some countably infinite subset $X$ of $[1/2, 1]$ and for any given $n$, consider the subset $M_n$ say of $M$ comprising sets of the form $\{x_1, x_2, \ldots, x_n\} \cup X$, where $0 < x_1 < x_2 < \ldots < x_n < 1/2$. Then (for any reasonable notion of distance between two sets) $M_n$ is an $n$-dimensional manifold topologically embedded in $M$. Since this holds for every $n$, $M$ cannot be an $n$-dimensional manifold for any $n$ (by invariance of domain).