PROBLEM: Let $M$ be a set. Let $\mathcal{A}=\{(U_{\alpha}, x_{\alpha})\}$ be an $\mathbb{R}^n$-valued atlas of class $C^r$ on $M$. Recall that an $\mathbb{R}^n$-valued chart $(V,y)$ is compatible with $\mathcal{A}$ if $\mathcal{A}\cup \{(V,y)\}$ is an $\mathbb{R}^n$-valued atlas of class $C^r$ on $M$. Let each of the $\mathbb{R}^n$-valued charts $(V,y),(W,z)$ be compatible with $\mathcal{A}$. Show that $(V,y)$ and $(W,z)$ are $C^r$-related charts.
EDIT: Recall, a pair of charts are $C^r$-related if either $V\cap W=\emptyset$ or both $y(V\cap W)$ and $z(V\cap W)$ are open and $y\circ z^{-1}$ and $z\circ y^{-1}$ are $C^r$ maps (r times differentiable)
Def: Let $M$ be a set. A chart on $M$ is a bijection of a subset $U\subset M$ onto an open subset of some euclidean space $\mathbb{R}^n$
Note that $M$ is just a set so we can't assume an induced topology. Also, we can't assume that the charts are homeomorphisms. When I attempted this problem I got it completely wrong because I assumed that the charts were homeomorphisms which is how some texts define a chart.