Classification of Compact One-Manifolds

Theorem
Every compact connected one-dimensional manifold is diffeomorphic to either a circle or a closed interval.

Lemma 1
Let $f$ be a Morse function on a one-manifold $X$.

Let $S$ be the union of the critical points of $f$ and $\partial X$.

As $S$ is finite, $X \setminus S$ consists of a finite number of one-manifolds, $L_1, L_2, \cdots, L_n$.