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.