Classification of Compact One-Manifolds/Lemma 3

Lemma for Classification of Compact One-Manifolds

Let $L$ be a subset of $X$ diffeomorphic to an open interval in $\R$, where $\dim X = 1$.

Then the closure $\map \cl L$ contains at most two points not in $L$.

Proof

Let $g$ be a diffeomorphism:

$g: \openint a b \to L$

Let $p \in \map \cl L \setminus L$.

Let $J$ be a closed subset of $X$ diffeomorphic to $\closedint 0 1$ such that:

$1$ corresponds to $p$
$0$ corresponds to some $\map g t$ in $L$.

Consider the set $\set {s \in \openint a t: \map g s \in J}$.

This set is both open and closed in $\openint a b$.

Hence $J$ contains either $g \sqbrk {\openint a t}$ or $g \sqbrk {\openint t b}$.

$\blacksquare$