Classification of Compact One-Manifolds/Lemma 2

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Lemma for Classification of Compact One-Manifolds

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

$f$ maps each $L_i$ diffeomorphically onto an open interval in $\R$.

Proof

Let $L$ be any of the $L_i$.

Because $f$ is a local diffeomorphism and $L$ is connected, $f \sqbrk L$ is open and connected in $\R$.

We also have $f \sqbrk L \in f \sqbrk X$, the latter of which is compact.

Hence there are numbers $c$ and $d$ such that $f \sqbrk L = \openint c d$.

It suffices to show $f$ is one to one on $L$, because then $f^{-1}: \openint c d \to L$ is defined and locally smooth.

Let $p$ be any point of $L$.

Set $q = \map f p$.

It suffices to show that every other point $z \in L$ can be joined to $p$ by a curve $\gamma: \closedint q y \to L$ such that $f \circ \gamma$ is the identity and $\map \gamma y = z$.

Since $\map f z = y \ne q = \map f p$, this result shows $f$ is one to one.

So let $Q$ be the set of points $x$ that can be so joined.

Since $f$ is a local diffeomorphism, $Q$ is both open and Definition:Closed Set (Topology).

Hence $Q = L$.

$\blacksquare$

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