In complex analysis, a field in mathematics, the RemmertâÂÂStein theorem, introduced by , gives conditions for the closure of an analytic set to be analytic.
The theorem states that if F is an analytic set of dimension less than k in some complex manifold D, and M is an analytic subset of D â 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 is analytic in the complex plane minus the origin, but its closure in the complex plane is not.
A consequence of the RemmertâÂÂStein theorem (also treated in their paper), is Chow's theorem stating that any projective complex analytic space is necessarily a projective algebraic variety.
The RemmertâÂÂStein theorem is implied by a proper mapping theorem due to , see .