This might be something well-known.

Let $A(n,k)\subset\mathbb{R}^{n}$ be the set of points $x=(x_{1},...,x_{n}% )$ with at least $k$ distinct coordinates. Then what are the homologies of $A(n,k)$?

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

This might be something well-known.

For $1\le k\le n$, let $A(n,k)\subset\mathbb{R}^{n}$ be the set of points $x=(x_{1},...,x_{n}% )$ with at least $k$ distinct coordinates. Then what are the homologies of $A(n,k)$?

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

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 $\mathbb{S}^{n-2}$.

This might be something well-known.

Let $A(n,k)\subset\mathbb{R}^{n}$ be the set of points $x=(x_{1},...,x_{n}% )$ with at least $k$ distinct coordinates. Then what are the homologies of $A(n,k)$?

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