PlanetMath: subanalytic set

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subanalytic set

(Definition)

Let $U \subset {\mathbb{R}}^n$ . Suppose $\mathcal{A}(U)$ is any ring of real valued functions on . Define $\mathcal{S}(\mathcal{A}(U))$ to be the smallest set of subsets of , 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 1 A set $V \subset {\mathbb{R}}^n$ is semianalytic if and only if for each $x \in {\mathbb{R}}^n$ , there exists a neighbourhood

, 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 2 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

, such that $V \cap U$ 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 $\dim_p A$ is the dimension of the submanifold. The dimension of the subanalytic set is the maximum $\dim_p A$ 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 1 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 3 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

(i.e. the set $\{ (x,y) \in U \times {\mathbb{R}}^m ~\mid~ x, y=f(x) \}$ ) is subanalytic (resp. semianalytic)

Bibliography

1: Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5-42. MR 89k:32011

"subanalytic set" is owned by jirka.

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Also defines:

subanalytic, semianalytic set, semianalytic, semianalytic function, subanalytic function, semianalytic mapping, subanalytic mapping, dimension of a subanalytic set

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Cross-references: graph, mapping, locally finite, subvariety, contained, point, dimension, submanifolds, dense set, open, coordinates, onto, relatively compact, projections, Tarski-Seidenberg theorem, semialgebraic sets, neighbourhood, complement, intersection, union, finite, closed under, contain, subsets, functions, real, ring
There are 2 references to this entry.

This is version 3 of subanalytic set, born on 2007-02-28, modified 2007-02-28.
Object id is 8999, canonical name is SubanalyticSet.
Accessed 2213 times total.

Classification:

AMS MSC:	32B20 (Several complex variables and analytic spaces :: Local analytic geometry :: Semi-analytic sets and subanalytic sets)
	14P15 (Algebraic geometry :: Real algebraic and real analytic geometry :: Real analytic and semianalytic sets)

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