# Subanalytic set

In mathematics, particularly in the subfield of real analytic geometry, a **subanalytic set** is a set of points (for example in Euclidean space) defined in a way broader than for **semianalytic sets** (roughly speaking, those satisfying conditions requiring certain real power series to be positive there). Subanalytic sets still have a reasonable local description in terms of submanifolds.

## Formal definitions

A subset *V* of a given Euclidean space *E* is **semianalytic** if each point has a neighbourhood *U* in *E* such that the intersection of *V* and *U* lies in the Boolean algebra of sets generated by subsets defined by inequalities *f* > 0, where f is a real analytic function. There is no Tarski–Seidenberg theorem for semianalytic sets, and projections of semianalytic sets are in general not semianalytic.

A subset *V* of *E* is a **subanalytic set** if for each point there exists a relatively compact semianalytic set *X* in a Euclidean space *F* of dimension at least as great as *E*, and a neighbourhood *U* in *E*, such that the intersection of *V* and *U* is a linear projection of *X* into *E* from *F*.

In particular all semianalytic sets are subanalytic. On an open dense subset, subanalytic sets are submanifolds and so they have a definite dimension "at most points". 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. On the other hand there is a theorem, to the effect that a subanalytic set *A* can be written as a locally finite union of submanifolds.

Subanalytic sets are not closed under projections, however, because 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.

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## External links

*This article incorporates material from Subanalytic set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*