Classification of Compact One-Manifolds/Corollary

Corollary to Classification of Compact One-Manifolds

A compact one-manifold has an even number of points in its boundary.

Proof

Let $M$ be a one-manifold. That is, $M$ is a topological manifold of dimension $1$.

By Classification of Compact One-Manifolds: Every compact connected one-dimensional manifold is diffeomorphic to either a circle or a closed interval.

It remains to be shown that a circle has an even number of points in its boundary.

It remains to be shown that a closed interval has an even number of points in its boundary.

By definition, the boundary of $M$ consists of all the points in the closure of $M$ which are not in the interior of $M$.

Thus, the boundary of $M$ is defined as:

$\partial M := M^- \setminus M^\circ$

where $M^-$ denotes the closure and $M^\circ$ the interior of $M$.

Hence a circle has an even number of points in its boundary.

For the case of the real interval,let:

$M = \closedint a b$

such that $a \ne b$, since it is a one-manifold.

By Closure of Real Interval is Closed Real Interval, it follows that the closure of $M$, $M^-$, is:

$\closedint a b$.

By Interior of Closed Real Interval is Open Real Interval, it follows that the interior of $M$, $M^\circ$ is the open interval:

$\openint a b$.

It follows that $a$, $b \in \partial M$. If:

$a \ne b$

then the number of points in its boundary is 2, which is an even number.

Hence a closed interval has an even number of points in its boundary.

Hence, a compact one-manifold has an even number of points in its boundary.

$\blacksquare$