subanalytic set
Let .
Suppose is any ring of real valued functions on
.
Define to be the smallest
set of subsets of , which contain the sets
for all ,
and is closed under finite union, finite intersection and complement.
Definition.
A set is semianalytic if and only if for each , there exists a neighbourhood of , such that , where denotes the real-analytic real valued functions.
Unlike for semialgebraic sets, there is no Tarski-Seidenberg theorem for semianalytic sets, and projections
of semianalytic sets are in general not semianalytic.
Definition.
We say
is a subanalytic set if for
each , there exists a relatively compact semianalytic set
and a neighbourhood of , such that
is the projection of onto the first coordinates.
In particular all semianalytic sets are subanalytic.
On an open dense set
subanalytic sets are submanifolds and hence we
can define dimension. Hence at a point , where a set is a submanifold,
the dimension is the dimension of the submanifold. The dimension of the subanalytic set is the maximum for all
where is a submanifold.
Semianalytic sets are contained in a real-analytic subvariety
of the same dimension. However, subanalytic sets are not in general
contained in any subvariety of the same dimension. We do have however the
following.
Theorem.
A subanalytic set can be written as a locally finite union of
submanifolds.
The set of subanalytic sets is still not completely closed under projections however. Note that a real-analytic subvariety that is not relatively compact can have a projection which is not a locally finite union of submanifolds, and hence is not subanalytic.
Definition.
Let . A mapping is said to be subanalytic (resp. semianalytic) if the graph of (i.e. the set ) is subanalytic (resp. semianalytic)
References
- 1 Edward Bierstone and Pierreย D. Milman, Semianalytic and subanalytic sets, Inst. Hautes รtudes Sci. Publ. Math. (1988), no.ย 67, 5โ42. http://www.ams.org/mathscinet-getitem?mr=89k:32011MR 89k:32011
Title | subanalytic set |
Canonical name | SubanalyticSet |
Date of creation | 2013-03-22 16:46:16 |
Last modified on | 2013-03-22 16:46:16 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 6 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 32B20 |
Classification | msc 14P15 |
Related topic | TarskiSeidenbergTheorem |
Related topic | SemialgebraicSet |
Defines | subanalytic |
Defines | semianalytic set |
Defines | semianalytic |
Defines | semianalytic function |
Defines | subanalytic function |
Defines | semianalytic mapping |
Defines | subanalytic mapping |
Defines | dimension of a subanalytic set |