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subanalytic set
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(Definition)
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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 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 $U$ 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 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 $U$ 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 1 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 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 $f$ (i.e. the set $\{ (x,y) \in U \times {\mathbb{R}}^m ~\mid~ x, y=f(x) \}$ 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. MR 89k:32011
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"subanalytic set" is owned by jirka.
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See Also: Tarski-Seidenberg theorem, semialgebraic set
| 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 3742 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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Pending Errata and Addenda
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