subanalytic set

Let $U\subset{\mathbb{R}}^{n}$. Suppose $\mathcal{A}(U)$ is any ring of real valued functions on $U$. Define $\mathcal{S}(\mathcal{A}(U))$ to be the smallest set of subsets of $U$, which contain the sets $\{x\in U\mid f(x)>0\}$ for all $f\in\mathcal{A}(U)$, and is closed under finite union, finite intersection and complement.

Definition.

A set $V\subset{\mathbb{R}}^{n}$ is semianalytic if and only if for each $x\in{\mathbb{R}}^{n}$, there exists a neighbourhood $U$ of $x$, such that $V\cap U\in\mathcal{S}(\mathcal{O}(U))$, where $\mathcal{O}(U)$ 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 $V\subset{\mathbb{R}}^{n}$ is a subanalytic set if for each $x\in{\mathbb{R}}^{n}$, there exists a relatively compact semianalytic set $X\subset{\mathbb{R}}^{n+m}$ and a neighbourhood $U$ of $x$, such that $V\cap U$ is the projection of $X$ onto the first $n$ 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 $p$, where a set $A$ is a submanifold, the dimension $\dim_{p}A$ is the dimension of the submanifold. The dimension of the subanalytic set is the maximum $\dim_{p}A$ for all $p$ where $A$ 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 $A$ 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 $U\subset{\mathbb{R}}^{n}$. A mapping $f\colon U\to{\mathbb{R}}^{m}$ is said to be subanalytic (resp. semianalytic) if the graph of $f$ (i.e. the set $\{(x,y)\in U\times{\mathbb{R}}^{m}~{}\mid~{}x,y=f(x)\}$) is subanalytic (resp. semianalytic)

References

• 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
 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