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'''Gromov's compactness theorem''' can refer to either of two mathematical theorems: * [[Gromov's compactness theorem (geometry)]] stating that certain sets of [[Riemannian manifold]]s ...

*One of Gromov's compactness theorems: ** [[Gromov's compactness theorem (geometry)]] in Riemannian geometry ...

...ompactness theorems, because they were formulated in terms of [[sequential compactness]] (the possibility of selecting a convergent subsequence). '''Mahler's compactness theorem''' states that a subset ''Y'' of ''X'' is [[relatively compact]] [[ ...

{{distinguish|text = [[Gromov's compactness theorem (geometry)|Gromov's compactness theorem in metric geometry]]}} ...d its generalizations to punctured pseudoholomorphic curves, underlies the compactness results for flow lines in [[Floer homology]] and [[symplectic field theory] ...

...equence]]s may be used to test for compactness, the criterion for relative compactness becomes that any sequence in {{mvar|Y}} has a subsequence convergent in {{m ..., and the concept of [[normal family]] in [[complex analysis]]. [[Mahler's compactness theorem]] in the [[geometry of numbers]] characterizes relatively compact s ...

Several theorems are named after [[Karl Weierstrass]]. These include: *The [[Bolzano–Weierstrass theorem]], which ensures compactness of closed and bounded sets in '''R'''^''n'' ...

{{short description|Relates tightness of measures to relative compactness in the space of probability measures}} ...'''Prokhorov's theorem''' relates [[tightness of measures]] to relative [[compactness]] (and hence [[Weak convergence of measures|weak convergence]]) in the spac ...

...d after [[Richard Palais]] and [[Stephen Smale]], is a hypothesis for some theorems of the [[calculus of variations]]. It is useful for guaranteeing the exist ...nction space]]s, the condition is necessary because some extra notion of [[compactness]] beyond simple boundedness is needed. See, for example, the proof of the ...

{{short description|Two theorems about families of holomorphic functions}} The conditions in the above theorems are sufficient, but not necessary for normality. Indeed, ...

{{distinguish|text = [[Gromov's compactness theorem (topology)|Gromov's compactness theorem in symplectic geometry]]}} ...automatically satisfied under an assumption on [[Ricci curvature]]. These theorems have been widely used in the fields of [[geometric group theory]] and [[Rie ...

== Theorems and specific laws == * [[Compactness theorem]] [[Talk:compactness theorem| ]] ...

==Theorems of which articles are primarily devoted to proving them== ==Articles devoted to theorems of which a (sketch of a) proof is given== ...

===Compactness and countability=== **[[Measure of non-compactness]] ...

...subsequent theories about the limits of provability|Gödel's incompleteness theorems}} ==Relationship to the incompleteness theorems== ...

Useful theorems: [[Category:Compactness (mathematics)]] ...

=== From the compactness theorem === ...larger than any numeral mentioned in the finite subset of P*. Thus by the compactness theorem there is a model satisfying all the axioms P*. Since any model of P ...

...^''n'').</ref> The theorem is sometimes called the '''sequential compactness theorem'''.<ref>Fitzpatrick 2006, p. xiv.</ref> == Sequential compactness in Euclidean spaces == ...

{{distinguish|Mahler's compactness theorem}} [[Category:Theorems in mathematical analysis]] ...

{{For|other theorems named after Tychonoff|Tychonoff's theorem (disambiguation)}} ...theorem depends crucially upon the precise definitions of [[compact space|compactness]] and of the [[product topology]]; in fact, Tychonoff's 1935 paper defines ...

| name = Compactness theorem In [[mathematical logic]], the '''compactness theorem''' states that a [[Set (mathematics)|set]] of [[First-order predica ...