Submanifold: Difference between revisions

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→‎Immersed submanifolds: need not be one-to-one
(→‎Immersed submanifolds: need not be one-to-one)
[[Image:immersedsubmanifold_nonselfintersection.jpg|thumb|150px|Immersed submanifold open interval with interval ends mapped to arrow marked ends.]]
An '''immersed submanifold''' of a manifold ''M'' is the image ''S'' of an [[immersion (mathematics)|immersion]] map ''i'': ''N'' → ''M''; in general this image will not be a submanifold as a subset, and an immersion map need not even be one-to-one – it can have self-intersections – so the term is used loosely.
More narrowly, one can require that the map ''i'': ''N'' → ''M'' be an inclusion (one-to-one), in which we call it an [[injective]] [[immersion (mathematics)|immersion]], and define an immersed submanifold to be the image subset ''S'' together with a [[topology (structure)|topology]] and [[differential structure]] such that ''S'' is a manifold and the inclusion ''i'' is a [[diffeomorphism]]: this is just the topology on ''N,'' which in general will not agree with the subset topology: in general the subset ''S'' is not a submanifold of ''M,'' in the subset topology.