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.

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.

On an open dense subset, subanalytic sets are submanifolds and so they have a definite dimension "at most points".

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.