A Manifold which can be ``charted'' with finitely many Euclidean Space charts. The Circle is the only compact 1-D Manifold. The Sphere and -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 of dimensions.
See also Tychonof Compactness Theorem