Classification of finite-dimensional universal pseudo-boundaries and pseudo-interiors

Let $n$ and $k$ be fixed integers such that $n \geq 1$ and $0 \leq k \leq n$. Let $B_k^n$ and $s_k^n$ denote the $k$-dimensional universal pseudo-boundary and the $k$-dimensional universal pseudo-interior in ${{\mathbf{R}}^n}$, respectively. The aim of this paper is to prove that $B_k^n$ is homeomorphic to $B_k^m$ if and only if $s_k^n$ is homeomorphic to $s_k^m$ if and only if $n = m$ or $n$, $m \geq 2k + 1$.