This might be something well-known.

Let $A(n,k)$ denote the family of finite subsets $M\subset\mathbb{R}$ of
cardinality $k\leq|M|\leq n$. Then what are the homologies of $A(n,k)$? Here
$A(n,k)$ is equipped with the Hausdorff topology.

For example, $A(n,2)$ is homeomorphic to $\mathbb{R}^{n}\backslash\Delta$,
where $\Delta$ is the diagonal, thus it is homotopy equivalent to
$\mathbb{S}^{n-2}$.