The Whitney (or strong) topology is a topology assigned to the space of mappings from a manifold to a manifold having continuous derivatives . It gives a notion of proximity of two mappings, and it allows us to speak of “robustness” of properties of a mapping. For example, the property of being an embedding is robust: if is a embedding, then there is a strong neighborhood of in which any mapping is an embedding.
as the set of mappings such that for all we have and
That is, those maps that are close to and have their first derivatives close to the respective first -th derivatives of , in local coordinates. It can be checked that the set of all such neighborhoods forms a basis for a topology, which we call the Whitney or strong topology of .
The weak topology, or compact-open topology, is defined in the same fashion but instead of choosing to be a locally finite atlas for , we require it to be an arbitrary finite family of charts (possibly not covering ).
The space with the weak or strong topologies is denoted by and , respectively.
We have that is always metrizable (with a complete metric) and separable. On the other hand, is not even first countable (thus, not metrizable) when is not compact; however, it is a Baire space. When is compact, the weak and strong topologies coincide.
|Date of creation||2013-03-22 14:08:27|
|Last modified on||2013-03-22 14:08:27|
|Last modified by||Koro (127)|