I am following D.gale's outline of proof for the classification theorem.
(http://www.igt.uni-stuttgart.de/eiserm/lehre/2014/Topologie/Gale%20-%201-manifolds.pdf)
For two charts $(U,\phi)$ and $(V,\psi)$ of a 1-manifold $X$
(which meets but not included in each other),
I have these two propositions
1. Suppose $X$ is compact and connected.
if $X=U\cup V$ and $U \cap V$ has two connected components, then $X$ is homeomorphic to a circle $C^1$.
2. If $X=U \cap V$ is connected, $U \cup V$ is homeomorphic to $\mathbb R$.
(Plus, It is also given that $U \cap V$ has at most two components.)
I am trying to prove that every compact connected 1-manifold $X$ is homeomorphic to $C^1$ by generalizing the first proposition.
To be concrete, I took a finite open cover $\mathcal O$ of $X$ by its subsets which are homeomorphic to $\mathbb R$ which has no proper subcover.
As Gale suggested in his text, I tried to use induction on $\vert \mathcal O \vert$.
Proposition 1 covers the case where $\vert \mathcal O \vert = 2$, and I am pretty sure that I would have to use Proposition 2 to make the $k \to k+1$ argument.
However, I cannot show that there is at least one pair $(U_i,U_j)$ of elements in $\mathcal O$ such that $U_i \cap U_j$ is nonempty and connected, which seems necessary to apply the Propositon 2 to this argument.
How can I get around with this problem?
Thanks in advance.
Tl;dr : Help me in showing the compact case of classification theorem for 1-manifolds without boundary, from the two propositions mentioned above.
(edit : typo)