- #1
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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.
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.