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.
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.
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.
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.
Let . A mapping is said to be subanalytic (resp. semianalytic) if the graph of (i.e. the set ) is subanalytic (resp. semianalytic)
- 1 Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42. http://www.ams.org/mathscinet-getitem?mr=89k:32011MR 89k:32011
|Date of creation||2013-03-22 16:46:16|
|Last modified on||2013-03-22 16:46:16|
|Last modified by||jirka (4157)|
|Defines||dimension of a subanalytic set|