# compactness

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**Mumford's compactness theorem**— In mathematics, Mumford s compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metric is compact. It was …22

**Gromov's compactness theorem**— can refer to either of two mathematical theorems:* Gromov s compactness theorem (geometry) in Riemannian geometry * Gromov s compactness theorem (topology) in symplectic topology …23

**Gromov's compactness theorem (topology)**— For Gromov s compactness theorem in Riemannian geometry, see that article. In symplectic topology, Gromov s compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have …24

**Barwise compactness theorem**— In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first order logic to a certain class of infinitary languages. It was stated and proved by Barwise in… …25

**Palais-Smale compactness condition**— The Palais Smale compactness condition is a necessary condition for some theorems of the calculus of variations.The condition is necessary because the calculus of variations studies function spaces that are infinite dimensional some extra notion… …26

**Gromov's compactness theorem (geometry)**— For Gromov s compactness theorem in symplectic topology, see that article. In Riemannian geometry, Gromov s (pre)compactness theorem states that the set of Riemannian manifolds of a given dimension, with Ricci curvature ≥ c and diameter ≤ D is… …27

**sequential compactness**— noun The property of a metric space that every sequence has a convergent subsequence …28

**Compact space**— Compactness redirects here. For the concept in first order logic, see compactness theorem. In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness… …29

**Tychonoff's theorem**— For other theorems named after Tychonoff, see Tychonoff s theorem (disambiguation). In mathematics, Tychonoff s theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey… …30

**Limit point compact**— In mathematics, particularly topology, limit point compactness is a certain condition on a topological space which generalizes some features of compactness. In a metric space, limit point compactness, compactness, and sequential compactness are… …