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In complex analysis, a field in mathematics, the **Remmert–Stein theorem**, introduced by Reinhold Remmert and Karl Stein (1953), 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* – *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 Remmert–Stein theorem (also treated in their paper), is Chow's theorem stating that any projective complex analytic space is necessarily a projective algebraic variety.

The Remmert–Stein theorem is implied by a proper mapping theorem due to Bishop (1964), see Aguilar & Verjovsky (2021).

- Aguilar, Carlos Martínez; Verjovsky, Alberto (2021),
*Chow's Theorem Revisited*, arXiv:2101.09872 - Bishop, Errett (1964), "Conditions for the Analycity of certain sets",
*Michigan Math. J.*,**11**(4): 289–304, doi:10.1307/mmj/1028999180 - Kato, Kazuko (1966). "Sur le théorème de P. Thullen et K. Stein".
*Journal of the Mathematical Society of Japan*.**18**(2). doi:10.2969/jmsj/01820211. S2CID 122821030. - Remmert, Reinhold; Stein, Karl (1953), "Über die wesentlichen Singularitäten analytischer Mengen",
*Mathematische Annalen*,**126**: 263–306, doi:10.1007/BF01343164, ISSN 0025-5831, MR 0060033, S2CID 119966389

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