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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
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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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contained in any subvariety of the same dimension. On the other hand, there is a theorem, to the effect that a subanalytic set
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to be positive there). Subanalytic sets still have a reasonable local description in terms of
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for semianalytic sets, and projections of semianalytic sets are in general not semianalytic.
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This article incorporates material from
Subanalytic set on
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of sets generated by subsets defined by inequalities
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Real
Algebraic and Analytic Geometry Preprint Server
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