Math 2.0

Several complex variables and analytic spaces Local analytic geometry Theorem

Gabrielov's theorem of the complement

Builds on Subanalytic setSemianalytic setO-minimal theoryTarski–Seidenberg theorem

Statement

Let X⊆ℝnX \subseteq \R^n be a subanalytic set. Then the complement ℝn∖X\R^n \setminus X is subanalytic.

Remarks

For semianalytic sets the analogous fact is immediate, since the complement of a set defined by equations and inequalities is again so defined. A subanalytic set is locally a projection, and the complement of a projection is not visibly a projection. Overcoming this is the content of the theorem; Gabrielov's proof studies the fibres of projections of semianalytic sets. For example, if A⊆ℝn+mA \subseteq \R^{n+m} is bounded and semianalytic and π\pi forgets the last mm coordinates, then ℝn∖π(A)\R^n \setminus \pi(A) is subanalytic. With the theorem, the subanalytic subsets of ℝn\R^n form a Boolean algebra, and the closure, interior and boundary of a subanalytic set are subanalytic.

Going deeper

For a student

In semialgebraic geometry, closure under projection is the Tarski–Seidenberg theorem, and closure under complement is trivial. For subanalytic sets the roles are reversed: projections are built in, and complements are the hard part. Gabrielov's theorem is the starting point of the result that the real field with restricted analytic functions is an o-minimal structure.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Complete theory, concept, no evidence levelComplete theoryconceptField, concept, no evidence levelFieldconceptO-minimal theory, concept, no evidence levelO-minimal theoryconceptOrdered field, concept, no evidence levelOrdered fieldconceptPolynomials in several variables, concept, no evidence levelPolynomials in severa…conceptRational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectSemianalytic set, concept, no evidence levelSemianalytic setconceptSubanalytic set, concept, no evidence levelSubanalytic setconceptQuantifier elimination for real closed fields, statement, no evidence levelQuantifier eliminatio…statementReal closed field, example, no evidence levelReal closed fieldexampleTarski–Seidenberg theorem, statement, no evidence levelTarski–Seidenberg the…statementGabrielov's theorem of the complement, statement, no evidence levelGabrielov's theorem o…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complete theory concept
  2. Field concept
  3. O-minimal theory concept
  4. Ordered field concept
  5. Polynomials in several variables concept
  6. Rational numbers object
  7. Real numbers object
  8. Semianalytic set concept
  9. Subanalytic set concept
  10. Quantifier elimination for real closed fields statement
  11. Real closed field example
  12. Tarski–Seidenberg theorem statement
  13. Gabrielov's theorem of the complement statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Subanalytic_set
About this page
TagHH2EM3DPNK
URIm2://HH2EM3DPNK
Kindstatement, theorem
WritingW0 stub
MSC32B20, 14P15, 03C64
StatusThis copy of the site has no build data. Run m2 status HH2EM3DPNK.
AboutSubanalytic set
UsesSubanalytic set, Semianalytic set, O-minimal theory, Tarski–Seidenberg theorem

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