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.See also
* Semialgebraic setReferences
*Edward Bierstone and Pierre D. Milman, ''Semianalytic and subanalytic sets'', Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42.External links