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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

© 1996-9 Eric W. Weisstein