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theorem and states
** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.
Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.
has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.
For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.
For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.
* The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.
Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.
A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate ( i. e. ) the subalgebra generated by those elements is associative.
The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.
In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.
The no-hair theorem states that, once it achieves a stable condition after formation, a black hole has only three independent physical properties: mass, charge, and angular momentum.
For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.
The Cook – Levin theorem states that the Boolean satisfiability problem is NP-complete, and in fact, this was the first decision problem proved to be NP-complete.
Schaefer's dichotomy theorem states that, for any restriction to Boolean operators that can be used to form these subformulae, the corresponding satisfiability problem is in P or NP-complete.
Goldstone's theorem in quantum field theory states that in a system with broken continuous symmetry, there may exist excitations with arbitrarily low energy, called the Goldstone bosons.
In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.
Fermat's little theorem states that all prime numbers have the above property.
This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.
The Banach fixed point theorem states that a contraction mapping on a complete metric space admits a fixed point.
The fundamental theorem of calculus states that antidifferentiation is the same as integration.
The Nyquist – Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency of the signal ; but requires an infinite number of samples.
If X < sub > k </ sub > and Y < sub > k </ sub > are the DFTs of x < sub > n </ sub > and y < sub > n </ sub > respectively then the Plancherel theorem states:

theorem and topological
** Tychonoff's theorem stating that every product of compact topological spaces is compact.
His generalization of the classical Riemann-Roch theorem launched the study of algebraic and topological K-theory.
Then is a compact topological space ; this follows from the Tychonoff theorem.
Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.
One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem.
In topology, the Tietze extension theorem states that, if X is a normal topological space and
* Dylan G. L. Allegretti, Simplicial Sets and van Kampen's Theorem ( Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets ).
His other contributions include a simplified proof of the positive energy theorem involving spinors in general relativity, his work relating supersymmetry and Morse theory, his introduction of topological quantum field theory and related work on mirror symmetry, knot theory, twistor theory and D-branes and their intersections.
The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact.
They also include statements less obviously related to compactness, such as the De Bruijn – Erdős theorem stating that every minimal k-chromatic graph is finite, and the Curtis – Hedlund – Lyndon theorem providing a topological characterization of cellular automata.
The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem.
Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces.
In 1963, Vladimir Arnold discovered a topological proof of the Abel-Ruffini theorem, which served as a starting point for topological Galois theory.
The complement of a knot itself ( as a topological space ) is known to be a " complete invariant " of the knot by the Gordon – Luecke theorem in the sense that it distinguishes the given knot from all other knots up to ambient isotopy and mirror image.
In 1934, Lev Pontryagin proved the Pontryagin duality theorem ; a result on topological groups.
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space.
* Open mapping theorem ( topological groups ) states that a surjective continuous homomorphism of a locally compact Hausdorff group G onto a locally compact Hausdorff group H is an open mapping if G is σ-compact.
Like the open mapping theorem in functional analysis, the proof in the setting of topological groups uses the Baire category theorem.
The domain invariance theorem may be generalized to manifolds: if M and N are topological n-manifolds without boundary and f: M → N is a continuous map which is locally one-to-one ( meaning that every point in M has a neighborhood such that f restricted to this neighborhood is injective ), then f is an open map ( meaning that f ( U ) is open in N whenever U is an open subset of M ).
In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.

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