Inclusions of submflds. and diffeomorphisms.

[WWGD]

Hi, everyone:

I am a little confused about the issue of the inclusion map on submanifolds.

AFAIK, if S is a submanifold of M , then if we give S the weak (or initial) topology

of the inclusion, then the inclusion map is a homeomorphism.(his is the way I understand,

of making the inclusion map of a subspace into a space, into a homeomorphism. If

S is open in M, with this initial topology, then I think inclusion is also a diffeomorphism.)


Question: under what conditions on S is the inclusion map a diffeomorphism?.

I think that if S is closed in the topology of M, or at least not open in M , and

given the weak topology, then the inclusion may not be a diffeomorphism.

Is this correct?.

And when is a (topological) subspace S of M a submanifold, other than when S

is open in M, i.e., S is open as a subset of M?



Thanks.

 

[yyat]

You have to distinguish between immersed and embedded submanifolds (see also http://en.wikipedia.org/wiki/Submanifold" ). For immersed submanifolds the inclusion is an injective immersion, but not necessarily a homeomorphism onto its image (i.e. an embedding). If it is an embedding then the submanifold is an emdedded submanifold and the inclusion is also a diffeomorphism onto its image, this is not completely trivial and requires the implicit function theorem.

And when is a (topological) subspace S of M a submanifold, other than when S

is open in M, i.e., S is open as a subset of M?

If one can find charts of M such that S looks locally like R^k in R^n.