complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, a field in mathematics, the Remmert–Stein theorem, introduced by , gives conditions for the closure of an

analytic set In the mathematical field of descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student . Definition There are several equivalent d ...

to be analytic. The theorem states that if ''F'' is an analytic set of dimension less than ''k'' in some

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

''D'', and ''M'' is an analytic subset of ''D'' – ''F'' with all components of dimension at least ''k'', then the closure of ''M'' is either analytic or contains ''F''. The condition on the dimensions is necessary: for example, the set of points (1/''n'',0) in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

is analytic in the complex plane minus the origin, but its closure in the complex plane is not.

Relations to other theorems

A consequence of the Remmert–Stein theorem (also treated in their paper), is Chow's theorem stating that any projective complex

analytic space An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also a ...

is necessarily a

projective algebraic variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...

. The Remmert–Stein theorem is implied by a

proper mapping theorem Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...

due to , see .

References

* * * * Complex manifolds Theorems in complex analysis {{analysis-stub