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.