Submanifold: Difference between revisions

12 bytes added ,  3 years ago
→‎Submanifolds of Euclidean space: Euclidean space → real coordinate space (the latter is more accurately what is meant here; no metric is intended)
(→‎Embedded submanifolds: hyphen → en-dash)
(→‎Submanifolds of Euclidean space: Euclidean space → real coordinate space (the latter is more accurately what is meant here; no metric is intended))
Suppose ''S'' is an immersed submanifold of ''M''. If the inclusion map ''i'' : ''S'' → ''M'' is [[closed map|closed]] then ''S'' is actually an embedded submanifold of ''M''. Conversely, if ''S'' is an embedded submanifold which is also a [[closed subset]] then the inclusion map is closed. The inclusion map ''i'' : ''S'' → ''M'' is closed if and only if it is a [[proper map]] (i.e. inverse images of [[compact set]]s are compact). If ''i'' is closed then ''S'' is called a '''closed embedded submanifold''' of ''M''. Closed embedded submanifolds form the nicest class of submanifolds.
 
==Submanifolds of Euclideanreal coordinate space==
 
Manifolds are often ''defined'' as embedded submanifolds of [[Euclideanreal coordinate space]] '''R'''<sup>''n''</sup>, so this forms a very important special case. By the [[Whitney embedding theorem]] any [[second-countable space|second-countable]] smooth ''n''-manifold can be smoothly embedded in '''R'''<sup>2''n''</sup>.
 
==Notes==