Let $M$ be a topological space.
Definition. $M$ is locally Euclidean of dimension $n$ if each point of $M$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^n$.
More specifically, if $M$ is locally Euclidean of dimension $n$, for each $p\in M$ we can find
$(i)\;$ an open subset $U\subseteq M$ containing $p$,
$(ii)\;$ an open subset $\hat{U}\subseteq\mathbb{R}^n$, and
$(iii)\;$ a homeomorphism $\varphi\colon U\to\hat{U}.$
Give the following characterization for the locally Euclidean spaces of dimension $n$:
Proposition. A topological space $M$ is locally Euclidean of dimension $n$ iff either of the following properties holds:
$(a)\;$ Every point of $M$ has a neighborhood homeomorphic to an open ball in $\mathbb{R}^n.$
$(b)\;$ Every point of $M$ has a neighborhood homeomorphic to $\mathbb{R}^n$.
Proof. $(\Longleftarrow)$ If $M$ has the property $(a)$ or $(b)$, then $M$ is locally Euclidean of dimension $n$, (open balls are open in $\mathbb{R}^n$ and $\mathbb{R}^n$ is open).
$(\Longrightarrow)$ Suppose that $M$ is locally Euclidean of dimension $n$. Since any open ball in $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$ itself, properties $(a)$ and $(b)$ are equivalent, so we need only prove $(a)$. Let $p\in M$, for hypotesis exists a neighborhood $U$ of $p$ and a homeomorphism $\varphi\colon U\to\hat{U}$, where $\hat{U}\subseteq\mathbb{R}^n$ is open. Since $\hat{U}$ is open, exists $r>0$ such that $B:=B(\varphi(p),r)\subseteq\hat{U}$. Since $\varphi$ is continuous, $B$ is open in $\hat{U}$(because $B=\hat{U}\cap B$, and $B$ is open in $\mathbb{R}^n$) and $p\in\varphi^{-1}(B)$, the inverse image $\varphi^{-1}(B)\subseteq U$ is a neighborhood of $p$ ($\varphi^{-1}(B)$ is open in $U$).Therefore $\varphi_{|\varphi^-{1}(B)\times B}\colon\varphi^{-1}(B)\to B$ is the homeomorphism from a neighborhood of $p\in M$ and an open ball in $\mathbb{R}^n.$ $\square$
Question 1. Is the previous proof correct and sufficiently detailed?
At this point we know that if $M$ is locally Euclidean of dimension $n$, for each $p\in M$ we can find
$(i)\;$ an open subset $U\subseteq M$ containing $p$,
$(ii)\;$ an open balls $B:=B(\varphi(p),r)\subseteq\mathbb{R}^n$, and
$(iii)\;$ a homeomorphism $\varphi\colon U\to B.$
Now i want that $\varphi(p)=0$ and $r=1$.
Consider the map $\psi\colon B\to B(0,r)$, definied as $p\mapsto p-\varphi(p)$. It is easy to convince oneself that this map is a homeomorphism. Moreover, consider the map $f\colon B(0,r)\to B(0,1)$ defined as $p\mapsto p/r$, it is a homeomorphism. Then $f\circ\psi\circ\varphi\colon U\to B(0,1)$ is an homeomorphism. We can now rewrite the definition of locally Euclidean of dimension $n$ as follows: $M$ is locally Euclidean of dimension $n$, if for each $p\in M$ we can find
$(i)\;$ an open subset $U\subseteq M$ containing $p$,
$(ii)\;$ a homeomorphism $\varphi\colon U\to B(0,1).$
Question 2. Is this procedure correct? Is there any mathematical error in this procedure? Any comments?
Thanks!
Regarding question (1), it's correct.
Regarding question (2), you've made a mistake by writing the translation map $\psi$ as $p\mapsto p-\varphi(p)$. It should be $k\mapsto k-\varphi(p)$. The other parts are fine.