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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Subanalytic set</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In mathematics, particularly in the subfield of real analytic geometry, a <b>subanalytic set</b> is a set of points (for example in <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>) defined in a way broader than for <b>semianalytic sets</b> (roughly speaking, those satisfying conditions requiring certain real <a href="Power_series" title="Power series">power series</a> to be positive there). Subanalytic sets still have a reasonable local description in terms of <a href="Submanifold" title="Submanifold">submanifolds</a>.
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<div class="mw-heading mw-heading2"><h2 id="Formal_definitions">Formal definitions</h2></div>
<p>A subset <i>V</i> of a given Euclidean space <i>E</i> is <b>semianalytic</b> if each point has a neighbourhood <i>U</i> in <i>E</i> such that the intersection of <i>V</i> and <i>U</i> lies in the <a href="Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> of sets generated by subsets defined by inequalities <i>f</i> > 0, where f is a <a href="Real_analytic_function" class="mw-redirect" title="Real analytic function">real analytic function</a>. There is no <a href="Tarski%E2%80%93Seidenberg_theorem" title="Tarski–Seidenberg theorem">Tarski–Seidenberg theorem</a> for semianalytic sets, and projections of semianalytic sets are in general not semianalytic.
</p><p>A subset <i>V</i> of <i>E</i> is a <b>subanalytic set</b> if for each point there exists a <a href="Relatively_compact" class="mw-redirect" title="Relatively compact">relatively compact</a> semianalytic set <i>X</i> in a Euclidean space <i>F</i> of dimension at least as great as <i>E</i>, and a neighbourhood <i>U</i> in <i>E</i>, such that the intersection of <i>V</i> and <i>U</i> is a linear projection of <i>X</i> into <i>E</i> from <i>F</i>.
</p><p>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 <i>A</i> can be written as a <a href="Locally_finite_collection" title="Locally finite collection">locally finite</a> union of submanifolds.
</p><p>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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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Semialgebraic_set" title="Semialgebraic set">Semialgebraic set</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li>Edward Bierstone and Pierre D. Milman, <i>Semianalytic and subanalytic sets</i>, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42. <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0972342">0972342</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://www.maths.manchester.ac.uk/raag/"><i>Real Algebraic and Analytic Geometry Preprint Server</i></a></li></ul>
<p><i>This article incorporates material from Subanalytic set on <a href="PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>
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This article is issued from <a class="external text" title="Last edited on 2023-11-07" href="https://en.wikipedia.org/wiki/?title=Subanalytic_set&oldid=1183923033">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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