Compact Manifold

A Manifold which can be ``charted'' with finitely many Euclidean Space charts. The Circle is the only compact 1-D Manifold. The Sphere and $n$-Torus are the only compact 2-D Manifolds. It is an open question if the known compact Manifolds in 3-D are complete, and it is not even known what a complete list in 4-D should look like. The following terse table therefore summarizes current knowledge about the number of compact manifolds $N(D)$ of $D$ dimensions.

$D$	$N(D)$
1	1
2	2

See also Tychonof Compactness Theorem