Show $ M_1 \# M_2= U_1 \cup U_2 $ for some open subsets $U_i$ of manifolds $M_i$ such that the following properties hold. This is problem 4-19 of Lee's "Introdunction to topological manifolds" 2ed.
I want to show that for two connected $n$-manifolds $M_1$ and $M_2$, there are open subsets $U_1$ and $U_2$ of the connected sum $M_1 \# M_2$ and points $p_i \in M_i$ such that $U_i \cong M_i\setminus \{p_i\}$, $U_1 \cap U_2 \cong \mathbb R^n \setminus\{0\}$, and $U_1 \cup U_2 = M_1 \# M_2$ where an equivalence means homeomophism.
What I have tried:
By definition, $M_1 \# M_2$ is formed by attaching $M_1 \setminus B_1$ to $M_2 \setminus B_2$ along the boundary. I tried choosing $p_i$'s to be the center of the coordinate balls $B_i$. Then if the equivalences were homotopy equivalences, the statement seems to be true.
Any help is appreciated.
 A: Set up notation:
Let $B_i\subseteq M_i$ be open discs and $N_i$ normal neighbourhoods of the boundaries. Let $\varphi:N_1\to N_2$ a diffeomorphism that induces the gluing map (ie $\varphi$ maps the inward collar of $N_1$ to the outward collar of $N_2$ and vice versa). Let $V_i$ be the points of the open discs not in the normal neighbourhoods, note $V_i$ is a closed disk.
Then $M_1\#M_2$ is equal to $M_1 - V_1$ and $M_2-V_2$ glued along the normal collars, ie $M_1\#M_2$ is defined as the quotient space
$$(M_1- V_1) \cup_\varphi (M_2- V_2):= [(M_1-V_1)\cup (M_2-V_2)]/\{x\sim \varphi(x)\}$$
Let $U_i$ then be the image of $M_i- V_i$ in the quotient space. This is open since the quotient map necessarily is an open map. Further it is diffeomorphic to $M_i-V_i$ since this quotient map is a local diffeomorphism and it is injective on this set. The intersection $U_1\cap U_2$ is then the image of both normal neighbourhoods $N_1, N_2$ under the quotient mapping - infact these two neighbourhoods have the same image so it suffice to just take the image of $N_1$. But the quotient map is a local diffeomorphism and injective on $N_1$, hence $U_1\cap U_2\cong N_1$.
Now prove:

*

*For any connected manifold $M$ if you remove a closed disk the result is diffeomorphic to $M-\{p\}$ for a point $p$.

*A normal neighbourhood of the boundary of an $n$-dimensional ball is diffeomorphic to $\Bbb R^n - \{0\}$.

