Let $A$ be an atlas on the set $M$ and let $ x: U \to x(U) $ and $ y : V \to y(V )$ be bijections from subsets $U, V \subset M$ to open sets $x(U), y(V ) \subset \mathbb{R^n} $. Show that if the charts $(x,U)$ and $(y, V )$ are each related to every chart $(z,W) \in A$ , then $(x,U)$ is related to $(y, V )$ as well.
I can show that $x \circ y^{-1}$ and $y \circ x^{-1}$ are smooth (the first half of proof) but I am not quite sure how to prove that $x(U \cap V)$ and $y(U \cap V)$ are both open sets in $\mathbb{R^n}$.
I'd be thankful if someone can help me with this,
Let's show that $x$ is a homeomorphism (same goes for $y$). It's enough to show this locally. Let $p \in U$ and let $(\varphi, W) \in A$ with $p \in W$. Since $x = (x \circ \varphi^{-1}) \circ \varphi$ on $U \cap W$, and both $\varphi$ and $x \circ \varphi^{-1}$ are homeomorphisms (because $(x,U)$ is related to $(\varphi, W)$), this shows that $x$ is a homeomorphism on $U \cap W$.
So we have showed that for any $p \in U$, $x$ is a homeomorphism restricted to some neighborhood of $p$. In other words $x : U \to x(U)$ is a local homeomorphism. Since $x$ is bijective, it is a homeomorphism.
In particular, $x(U \cap V)$ is open because $U\cap V$ is open.
The same proof goes for $y$: $y$ is a homeomorphism so $y(U \cap V)$ is open.