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(→Embedded submanifolds: hyphen → endash) 
(→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 mapclosed]] 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
Manifolds are often ''defined'' as embedded submanifolds of [[
==Notes==
