The Classification of Connected Compact 1-Manifolds and 2-Manifolds

# The Classification of Connected Compact 1-Manifolds and 2-Manifolds

We will now state a series of results which classify all of the connected compact 1-manifolds and 2-manifolds (up to homeomorphism). We will not prove any of the results below as many of them are rather technical, though the results will be important for later.

## The Classification of Connected 1-Manifolds

 Theorem 1 (The Classification of Connected Compact 1-Manifolds): Every connected compact 1-manifold is homeomorphic to the circle \$S^1\$, and every connected noncompact 1-manifold is homeomorphic to \$\mathbb{R}\$.

Theorem 1 above tells us that every connected 1-manifold is homeomorphic to either \$S^1\$ or \$\mathbb{R}\$

## The Classification of Connected Compact 2-Manifolds

 Theorem 2 (The Classification of Connected Compact 2-Manifolds): Every connected compact 2-manifold is homeomorphic to either the sphere \$S^2\$, a connected sum of a finite number of toruses, or a connected sum of a finite number of projective planes.