# Homeomorphism type of pair of faces in a regular CW complex

Let $X$ be a regular CW complex, $\sigma$ an $n$-dimensional cell of $X$ and $\tau$ an $(n-1)$-dimensional face of $\sigma$. Is it true that the pair $(\bar\sigma, \bar\tau)$ is homeomorphic to the pair $(B^n, B^{n-1}_+)$?

Here $\bar\nu$ is the closure in $X$ of a cell $\nu$, $B^n$ is the closed $n$-ball $\{ \|x\| \leq 1 \} \subseteq \mathbb{R}^n$, and $B^{n-1}_+$ is the closed upper hemisphere of the boundary of $B^n$, i.e. $B^{n-1}_+ = \{ x \in \partial B^n \mid x_n \geq 0\}$.

Recall that a CW complex is regular if all attaching maps are homeomorphisms onto their image. In particular, if $\nu$ is an $n$-cell then $\bar \nu$ is homeomorphic to $B^n$.