In mathematics, particularly in the subfield of
real analytic geometry, 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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requiring certain real
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
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 S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
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.
See also
*
Semialgebraic set
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