In mathematics, particularly in the subfield of
real analytic geometry
Real may refer to:
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, a subanalytic set is a set of points (for example in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
) 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
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
s.
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
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
of sets generated by subsets defined by inequalities ''f'' > 0, where f is a
real analytic function. There is no
Tarski–Seidenberg theorem In mathematics, the Tarski–Seidenberg theorem states that a set in (''n'' + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto ''n''-dimensional space, and the resulting set is still defin ...
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
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sin ...
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 set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and ine ...
References
*Edward Bierstone and Pierre D. Milman, ''Semianalytic and subanalytic sets'', Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42.
External links
''Real Algebraic and Analytic Geometry Preprint Server''
{{PlanetMath attribution, id=8999, title=Subanalytic set
Real algebraic geometry