34,105
edits
(→Immersed submanifolds: elaborate) 

[[Image:immersedsubmanifold_nonselfintersection.jpgthumb150pxImmersed submanifold open interval with interval ends mapped to arrow marked ends.]]
An '''immersed submanifold''' of a manifold ''M'' is
More narrowly, one can require that the map ''i'': ''N'' → ''M'' be an inclusion (onetoone), 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.
Given any injective immersion ''f'' : ''N'' → ''M'' the [[image (mathematics)image]] of ''N'' in ''M'' can be uniquely given the structure of an immersed submanifold so that ''f'' : ''N'' → ''f''(''N'') is a [[diffeomorphism]]. It follows that immersed submanifolds are precisely the images of injective immersions.
