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Theorem in mathematical logic
the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an
Compactness_theorem
Product of any collection of compact topological spaces is compact
topological spaces based on fuzzy sets. The theorem depends crucially upon the precise definitions of compactness and of the product topology; in fact, Tychonoff's
Tychonoff's_theorem
Type of mathematical space
of Euclidean space, compactness is equivalent to being closed and bounded, by the Heine–Borel theorem. The property of compactness often allows local information
Compact_space
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem. The Bolzano–Weierstrass theorem is
Bolzano–Weierstrass_theorem
Area of mathematical logic
that both the completeness and compactness theorems were implicit in Skolem 1923...." [Dawson, J. W. (1993). "The compactness of first-order logic:from Gödel
Model_theory
Type of logical system
to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization
First-order_logic
Existence and cardinality of models of logical theories
Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order
Löwenheim–Skolem_theorem
mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a
Barwise_compactness_theorem
Compactness theorem in Yang–Mills theory
differential geometry and in particular Yang–Mills theory, Uhlenbeck's compactness theorem is a result about sequences of (weak Yang–Mills) connections with
Uhlenbeck's compactness theorem
Uhlenbeck's_compactness_theorem
On when a family of real, continuous functions has a uniformly convergent subsequence
equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators
Arzelà–Ascoli_theorem
Fundamental theorem in mathematical logic
then Tennenbaum's theorem shows that it has no recursive non-standard models. The completeness theorem and the compactness theorem are two cornerstones
Gödel's_completeness_theorem
Compact embedding theorem concerning Sobolev spaces
selection theorem, since one "selects" a convergent subsequence. (However, today the customary name is "compactness theorem", whereas "selection theorem" has
Rellich–Kondrachov_theorem
Theorem in functional analysis
This theorem is also called the Banach–Alaoglu theorem or the weak-* compactness theorem and it is commonly called simply the Alaoglu theorem. If X {\displaystyle
Banach–Alaoglu_theorem
Subset of Euclidean space is compact if and only if it is closed and bounded
pedagogical history of compactness". arXiv:1006.4131v1 [math.HO]. Bredon 2013, Ch I., Theorem 7.9. Bredon 2013, Ch I., After Theorem 8.9.: "That is why we
Heine–Borel_theorem
Theorem in measure theory
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures
Prokhorov's_theorem
Topics referred to by the same term
Gromov's compactness theorem can refer to either of two mathematical theorems: Gromov's compactness theorem (geometry) stating that certain sets of Riemannian
Gromov's_compactness_theorem
Topics referred to by the same term
Gromov's theorem may mean one of a number of results of Mikhail Gromov: One of Gromov's compactness theorems: Gromov's compactness theorem (geometry)
Gromov's_theorem
On convergent subsequences of functions that are locally of bounded total variation
admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named
Helly's_selection_theorem
Gives conditions for a space of compact Riemann surfaces of genus > 1 to be compact
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
Mumford's_compactness_theorem
Theorem in symplectic topology
In the mathematical field of symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex
Gromov's compactness theorem (topology)
Gromov's_compactness_theorem_(topology)
Model of (first-order) Peano arithmetic that contains non-standard numbers
models of arithmetic can be demonstrated by an application of the compactness theorem. To do this, a set of axioms P* is defined in a language including
Non-standard model of arithmetic
Non-standard_model_of_arithmetic
Continuous real function on a closed interval has a maximum and a minimum
these definitions, continuous functions can be shown to preserve compactness: Theorem—If V , W {\displaystyle V,\ W} are topological spaces, f : V →
Extreme_value_theorem
Mathematical construction
include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization
Ultraproduct
Characterizes sets of lattices that are bounded in a certain sense
selection theorem, following an older convention used in naming compactness theorems, because they were formulated in terms of sequential compactness (the
Mahler's_compactness_theorem
Form of logic that allows quantification over predicates
his axiomatisation, Henkin proved that Gödel's completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order
Second-order_logic
Russian mathematician (born 1966)
noncollapsing theorem is that volume control is one of the preconditions of Hamilton's compactness theorem. As a consequence, Hamilton's compactness and the
Grigori_Perelman
Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical
List_of_theorems
Mathematical theorem in the study of analysis
Stone–Weierstrass theorem that weaken the assumption of local compactness. The Stone–Weierstrass theorem can be used to prove the following two statements, which
Stone–Weierstrass_theorem
On coloring infinite graphs
171: "It is straightforward to prove [the De Bruijn–Erdős theorem] using the compactness theorem for first-order logic" Rorabaugh, Tardif & Wehlau (2017)
De Bruijn–Erdős theorem (graph theory)
De_Bruijn–Erdős_theorem_(graph_theory)
Bounds the length of geodetic segments in Riemannian manifolds based in Ricci curvature
Gromov's compactness theorem (geometry) – On when a set of compact Riemannian manifolds of a given dimension is relatively compact Ambrose, W. A theorem of
Myers's_theorem
On when a set of compact Riemannian manifolds of a given dimension is relatively compact
fundamental compactness theorem for sequences of metric spaces. In the special case of Riemannian manifolds, the key assumption of his compactness theorem is automatically
Gromov's compactness theorem (geometry)
Gromov's_compactness_theorem_(geometry)
Russian-French mathematician
Gromov's compactness theorem, stating that the set of compact Riemannian manifolds with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Type of large cardinal in set theory
\kappa } -compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185) A language Lκ,κ is said to satisfy the weak compactness theorem if whenever
Weakly_compact_cardinal
Subfield of mathematics
Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order
Mathematical_logic
On the existence of hyperplanes separating disjoint convex sets
compactness in the hypothesis cannot be relaxed; see an example in the section Counterexamples and uniqueness. This version of the separation theorem
Hyperplane_separation_theorem
On decreasing nested sequences of non-empty compact sets
sequences of non-empty compact sets. Theorem. Let S {\displaystyle S} be a topological space. A decreasing nested sequence of non-empty compact, closed subsets
Cantor's_intersection_theorem
subsequence. The Delta-compactness theorem is similar to the Banach–Alaoglu theorem for weak convergence but, unlike the Banach-Alaoglu theorem (in the non-separable
Delta-convergence
Relates three different kinds of weak compactness in a Banach space
Eberlein–Šmulian theorem (named after William Frederick Eberlein and Witold Lwowitsch Schmulian) is a result that relates three different kinds of weak compactness in
Eberlein–Šmulian_theorem
Fundamental result of mathematical logic
sequent". Herbrand structure Herbrand interpretation Herbrand universe Compactness theorem J. Herbrand: Recherches sur la théorie de la démonstration. Travaux
Herbrand's_theorem
Theorem in functional analysis
considering compact operators on infinite-dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially
Min-max_theorem
On volumes in complete Riemannian n-manifolds whose Ricci curvature has a lower bound
to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem. Let M {\displaystyle M} be a complete n-dimensional Riemannian manifold
Bishop–Gromov_inequality
integer rectifiable current whose boundary has finite mass. It is a deep theorem of Federer-Fleming that the boundary is then also an integral current.
Flat_convergence
Theorem in topology
set). It also requires compactness and convexity of the set. The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces,
Brouwer_fixed-point_theorem
Injective polynomial functions are bijective
{C} } . The proof thus uses model-theoretic principles such as the compactness theorem to prove an elementary statement about polynomials. The proof for
Ax–Grothendieck_theorem
Theorem on extension of bounded linear functionals
compactness theorem and to the Boolean prime ideal theorem) may be used instead. Hahn–Banach can also be proved using Tychonoff's theorem for compact
Hahn–Banach_theorem
Planar maps require at most four colors
This can also be seen as an immediate consequence of Kurt Gödel's compactness theorem for first-order logic, simply by expressing the colorability of an
Four_color_theorem
Subset of a topological space whose closure is compact
test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. Some major theorems characterize
Relatively_compact_subspace
Soundness theorem Gödel's completeness theorem Original proof of Gödel's completeness theorem Compactness theorem Löwenheim–Skolem theorem Skolem's paradox
List of mathematical logic topics
List_of_mathematical_logic_topics
Branch of logic
structures under finite model theory include the compactness theorem, Gödel's completeness theorem, and the method of ultraproducts for first-order logic
Finite_model_theory
Topics referred to by the same term
Compactness can refer to: Compact space, in topology Compact operator, in functional analysis Compactness theorem, in first-order logic Compactness measure
Compactness_(disambiguation)
Type of topological space in mathematics
homeomorphism) locally compact Hausdorff space X. This is shown using the Gelfand representation. The notion of local compactness is important in the study
Locally_compact_space
Theorem in geometric topology
their published paper made use of an incorrect version of Hamilton's compactness theorem for Ricci flow. Huai-Dong Cao and Xi-Ping Zhu published a paper in
Poincaré_conjecture
Statement about linear functionals and measures
representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named
Riesz–Markov–Kakutani representation theorem
Riesz–Markov–Kakutani_representation_theorem
Axiom of set theory
orthonormal basis. The Banach–Alaoglu theorem about compactness of sets of functionals. The Baire category theorem about complete metric spaces, and its
Axiom_of_choice
Branch of differential geometry
space. Gromov's compactness theorem. The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff
Riemannian_geometry
Theorem in complex analysis
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved
Runge's_theorem
Geometry concept
manifolds with sectional curvature ≥ k, as described by Gromov's compactness theorem. Alexandrov spaces with curvature ≥ k were introduced by the Russian
Alexandrov_space
American mathematician (1943–2024)
1995, Hamilton extended Jeff Cheeger's compactness theory for Riemannian manifolds to give a compactness theorem for sequences of Ricci flows.[H95a] Given
Richard_S._Hamilton
Topological space where every sequence has a convergent subsequence
notions of compactness and sequential compactness are equivalent (using the axiom of countable choice). However, there exist sequentially compact topological
Sequentially_compact_space
Gives condition for a set of functions to be relatively compact in an Lp space
0)=u_{0}(x).} Arzelà–Ascoli theorem Helly's selection theorem Rellich–Kondrachov theorem Sudakov, V. N. (1957). "Criteria of compactness in function spaces".
Fréchet–Kolmogorov_theorem
Topics referred to by the same term
Stone–Weierstrass theorem The Bolzano–Weierstrass theorem, which ensures compactness of closed and bounded sets in Rn The Weierstrass extreme value theorem, which
Weierstrass_theorem
Cantor's theorem Cantor–Bernstein–Schroeder theorem Cayley's formula Cayley's theorem Clique problem (to do) Compactness theorem (very compact proof) Erdős–Ko–Rado
List_of_mathematical_proofs
Sufficient criterion for uniform convergence
field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function
Dini's_theorem
Result about when a matrix can be diagonalized
In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented
Spectral_theorem
On when a space equals the closed convex hull of its extreme points
called quasicompactness or convex compactness. Compactness implies convex compactness because a topological space is compact if and only if every family of
Krein–Milman_theorem
Fixed-point theorem for set-valued functions
theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset
Kakutani_fixed-point_theorem
Type of continuous linear operator
is therefore relatively compact by Montel's theorem. The compactness comes from viewing holomorphic functions only on compact subsets strictly inside
Compact_operator
German mathematician (1903–1988)
measure Mahler polynomial Mahler volume Mahler's theorem Mahler's compactness theorem Skolem–Mahler–Lech theorem Coates, J. H.; Van Der Poorten, A. J. (1994)
Kurt_Mahler
Theorem in topology
non-empty compact set. The result also holds if one works with the ball measure of non-compactness or the separation measure of non-compactness, since these
Kuratowski's intersection theorem
Kuratowski's_intersection_theorem
Mathematical result in differential geometry
Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the
Atiyah–Singer_index_theorem
Two theorems about families of holomorphic functions
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after
Montel's_theorem
Theorem about the intersections of d-dimensional convex sets
Radon's theorem as in the proof by Radon (1921). The infinite version then follows by the finite intersection property characterization of compactness: a collection
Helly's_theorem
Relation between genus, degree, and dimension of function spaces over surfaces
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension
Riemann–Roch_theorem
Theorem in differential geometry
Euler characteristic. Compactness of the surface is of crucial importance. Consider for instance the open unit disc, a non-compact Riemann surface without
Gauss–Bonnet_theorem
Complete manifolds of non-negative sectional curvature largely reduce to the compact case
non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll
Soul_theorem
Theorem about inclusions between Sobolev spaces
prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly
Sobolev_inequality
Theorem in harmonic analysis
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel
Plancherel_theorem
Theorem that any three objects in space can be simultaneously bisected by a plane
mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space
Ham_sandwich_theorem
Mathematical theorem
x) ≥ 0. To show compactness, show that the image of the unit ball of L2[a,b] under TK is equicontinuous and apply Ascoli's theorem, to show that the
Mercer's_theorem
Logical quantifier
persweb.wabash.edu. Retrieved 2019-12-14. This is a consequence of the compactness theorem. Kleene, Stephen (1952). Introduction to Metamathematics. Ishi Press
Uniqueness_quantification
Symplectic topology tool
interest; solutions are known as pseudoholomorphic curves. The Gromov compactness theorem is then used to show that the counts of flow lines defining the differential
Floer_homology
Extension of the Brouwer fixed-point theorem
The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces, which may be of infinite
Schauder_fixed-point_theorem
Mathematical logician and philosopher
theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two
Kurt_Gödel
On convergent subsequences of regulated functions
is well known that BV([0, T]; X) satisfies the compactness theorem known as Helly's selection theorem: given any sequence of functions (fn)n∈N in BV([0
Fraňková–Helly selection theorem
Fraňková–Helly_selection_theorem
Existence and uniqueness of solutions to initial value problems
known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard,
Picard–Lindelöf_theorem
Group in which each element has finite order
this infinite disjunction by using an infinite set of axioms: the compactness theorem implies that no set of first-order formulae can characterize the
Torsion_group
Sequences of convex sets in a bounded set have convergent subsequences
metric. The theorem is named for Wilhelm Blaschke. A succinct statement of the theorem is that the metric space of convex bodies is locally compact. Using
Blaschke_selection_theorem
Theorem in mathematics
reflexivity is equivalent to the weak compactness of the unit sphere, Victor L. Klee reformulated this as a compactness criterion for the unit sphere in 1962
James's_theorem
Basic result in harmonic analysis on compact topological groups
mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily
Peter–Weyl_theorem
Singularity theorem in Yang–Mills theory
^{1}(B^{4},\operatorname {Ad} ({\overline {\eta }}))} . Uhlenbeck's compactness theorem, also first published in the same journal Uhlenbeck, Karen (February
Uhlenbeck's singularity theorem
Uhlenbeck's_singularity_theorem
Set of sentences in a formal language
axiom schemes for this symbol. Compactness theorem Consistent set Deduction theorem Lindenbaum's lemma Löwenheim–Skolem theorem One way to specify a theory
Theory_(mathematical_logic)
Theorem relating continuity to graphs
continuous. The converse is true when Y {\displaystyle Y} is compact. (Note that compactness and Hausdorffness do not imply each other.) Proof First part:
Closed_graph_theorem
Mapping theorem in topology
mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X {\displaystyle
Lefschetz_fixed-point_theorem
Periodic set of points
cryptography Lattice graph Lattice (module) Lattice (order) Mahler's compactness theorem Reciprocal lattice Unimodular lattice Gruber, Peter M.; Lekkerkerker
Lattice_(group)
graph Logic gate Boolean analysis Boolean prime ideal theorem Compactness theorem Consensus theorem De Morgan's laws Duality (order theory) Laws of classical
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Logical formulation of graph properties
zero–one law for first-order graph logic; Fagin's proof used the compactness theorem. According to this result, every first-order sentence is either almost
Logic_of_graphs
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Mathematical concept
supposed to play." In first-order logic, both the compactness theorem and Löwenheim–Skolem theorems are used to construct non-standard models with certain
Infinity
Class in model theory
pseudo-elementary class. Moreover, as an easy consequence of the compactness theorem, a class of σ-structures is basic elementary if and only if it is
Elementary_class
Partial differential equation
which leads to his "noncollapsing theorem". The noncollapsing theorem allows application of Hamilton's compactness theorem (Hamilton 1995) to construct "singularity
Ricci_flow
COMPACTNESS THEOREM
COMPACTNESS THEOREM
Girl/Female
Indian, Telugu
Completness
COMPACTNESS THEOREM
COMPACTNESS THEOREM
Girl/Female
American, Australian
Wisdom; Wise
Girl/Female
Tamil
Infinite, Endless, Eternal
Boy/Male
Muslim
Rightly guided
Girl/Female
Assamese, Hindu, Indian, Kannada, Marathi, Telugu
Writing; Studious
Boy/Male
Gujarati, Hindu, Indian, Kannada, Tamil
Tree of Knowledge; This is the Tree Where Buddha did Meditate and Gained Lot of Knowledge
Boy/Male
American, Anglo, Australian, British, Christian, English, French, Indian, Scottish, Welsh
Son of a Nobleman; Quick-moving; Speckled; Surname Form of Brice; Ardent; Strength; Pied; Spotted
Boy/Male
Hindu, Indian
Lovable
Boy/Male
Hindu, Indian, Telugu
Lord Indra
Girl/Female
Christian & English(British/American/Australian)
Maid of Battles
Male
English
Variant spelling of English Tyler, TYLOR means "roof-tiler."
COMPACTNESS THEOREM
COMPACTNESS THEOREM
COMPACTNESS THEOREM
COMPACTNESS THEOREM
COMPACTNESS THEOREM
n.
A numerical coefficient in any particular case of the binomial theorem.
n.
The quality or state of being trim; orderliness; compactness; snugness; neatness.
a.
Alt. of Theorematical
v. t.
To formulate into a theorem.
n.
A state of being compact.
n.
The quality of being dense, close, or thick; compactness; -- opposed to rarity.
n.
Thickness; density; compactness.
a.
Theorematic.
n.
The state or quality of being solid; density; consistency, -- opposed to fluidity; compactness; fullness of matter, -- opposed to openness or hollowness; strength; soundness, -- opposed to weakness or instability; the primary quality or affection of matter by which its particles exclude or resist all others; hardness; massiveness.
n.
The state or quality of being imporous; want of porosity; compactness.
n.
That which is considered and established as a principle; hence, sometimes, a rule.
n.
One who constructs theorems.
n.
A statement of a principle to be demonstrated.
n.
State or quality of being solid; firmness; compactness; solidity, as of material bodies.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
n.
The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.
n.
The state or quality of being compact; close union of parts; density.
n.
A theorem or proposition so easy of demonstration as to be almost self-evident.
a.
Containing many names or terms; multinominal; as, the polynomial theorem.
v. i.
To become hard or harder; to acquire solidity, or more compactness; as, mortar hardens by drying.