This is related to Hatcher Thm 2.26's proof.
If non empty open subsets $U\subset R^n, V\subset R^m$ are homeomorphic, then $m=n$.
Pf. $x\in U$, then $H_k(U,U-\{x\})=H_k(R^n,R^n-\{0\})$ which is $Z$ if $k=n$ and $0$ otherwise. Similarly $H_k(V,V-\{x\})=H_k(R^m,R^m-\{0\})$ which is $Z$ if $k=m$ and $0$ otherwise. Since $V\cong U$, one induces homology isomorphism which fixes $m=n$.
$\textbf{Q:}$ How did the book conclude for every open $U\subset R^n$, then $H_k(U,U-\{x\})=H_k(R^n,R^n-\{0\})$? I did not find this statement obvious. My reasoning is the following. Since $U$ is open, fix an open ball $B$ in $U$ and its image $B'$ contains a ball. WLOG, the statement specializes to open ball question and we can scale or translate the image in $R^m$ s.t. $B'$ contains unit ball centered at $0$ with the center $B$ at $0$ being sent to $0$. Denote map $f:B\to B'$. Then $f:B-\{0\}\to B'-\{0\}$ is homeomorphism. Now apply deformation retraction to $B'-\{0\}$ conclude $m=n$.
This follows from the excision axiom from the Eilenberg-Steenrod axioms for a homology theory. It states that if $(X,A)$ is a pair and $U \subset X$ is a subset such that the closure of $U$ is contained in the interior of $A$, then the inclusion from $(X \setminus U, A \setminus U)$ into $(X,A)$ induces an isomorphism on the homology groups.