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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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Subanalytic set」の詳細全文を読む スポンサード リンク
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