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Enyokoyama (talk  contribs) (→Embedded submanifolds: add links to "Alexander's theorem" and "Schoenflies theorem") 

There is an intrinsic definition of an embedded submanifold which is often useful. Let ''M'' be an ''n''dimensional manifold, and let ''k'' be an integer such that 0 ≤ ''k'' ≤ ''n''. A ''k''dimensional embedded submanifold of ''M'' is a subset ''S'' ⊂ ''M'' such that for every point ''p'' ∈ ''S'' there exists a [[chart (topology)chart]] (''U'' ⊂ ''M'', φ : ''U'' → '''R'''<sup>''n''</sup>) containing ''p'' such that φ(''S'' ∩ ''U'') is the intersection of a ''k''dimensional [[plane (mathematics)plane]] with φ(''U''). The pairs (''S'' ∩ ''U'', φ<sub>''S'' ∩ ''U''</sub>) form an [[atlas (topology)atlas]] for the differential structure on ''S''.
[[Alexander's theorem]] and the [[Schoenflies theoremJordanSchoenflies theorem]] are good examples of smooth embeddings.
===Other variations===
