topologies
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 βrobustnessβ
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.
Given a locally finite atlas and compact sets
such that there are charts
of for which
for all , and given a sequence
, we define the basic neighborhood
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.
Title | topologies |
---|---|
Canonical name | mathcalCrTopologies |
Date of creation | 2013-03-22 14:08:27 |
Last modified on | 2013-03-22 14:08:27 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 5 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 57R12 |
Synonym | Whitney topology |
Synonym | compact-open topology |
Synonym | weak topology |
Synonym | strong topology |
Related topic | ConjectureApproximationTheoremHoldsForWhitneyCrMNSpaces |
Related topic | ApproximationTheoremAppliedToWhitneyCrMNSpaces |