In dimension 3, Edwin E. Moise proved that every topological manifold has an essentially unique smooth structure ( see Moise's theorem ), so the monoid of smooth structures on the 3-sphere is trivial.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

** The Nielsen – Schreier theorem, that every subgroup of a free group is free.

** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals

** The Banach – Alaoglu theorem about compactness of sets of functionals.

** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.

This is a result of Galois theory ( see Quintic equations and the Abel – Ruffini theorem ).

Transmission, Gregory Chaitin also presents this theorem in J. ACM – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.

This began in his doctoral work leading to the Mordell – Weil theorem ( 1928, and shortly applied in Siegel's theorem on integral points ).

He had introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann – Roch theorem with them ( a version appeared in his Basic Number Theory in 1967 ).

His ' matrix divisor ' ( vector bundle avant la lettre ) Riemann – Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles.

* Borel – Weil theorem

* De Rham – Weil theorem

* Mordell – Weil theorem

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

** Well-ordering theorem: Every set can be well-ordered.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

In cybernetics, the Good Regulator or Conant-Ashby theorem is stated " Every Good Regulator of a system must be a model of that system ".

Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.

* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.

Every proper rotation is the composition of two reflections, a special case of the Cartan – Dieudonné theorem.

Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R < sup > 3 </ sup > which is called the axis of rotation ( this is Euler's rotation theorem ).

Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies.

Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection J from the definition is bijective, by the rank-nullity theorem.

Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.

Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof.

Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by

For example, to study the theorem “ Every bounded sequence of real numbers has a supremum ” it is necessary to use a base system which can speak of real numbers and sequences of real numbers.

# Every K3 surface is Kähler ( by a theorem of Y .- T. Siu ).

* Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem.

* Every automorphism of a central simple algebra is an inner automorphism ( follows from Skolem – Noether theorem ).

* Conway's cosmological theorem: Every sequence eventually splits into a sequence of " atomic elements ", which are finite subsequences that never again interact with their neighbors.

Every comparability graph is perfect: this is essentially just Mirsky's theorem, restated in graph-theoretic terms.

Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis ( Euler rotation theorem ).

Every classical theorem that applies to the natural numbers applies to the non-standard natural numbers.

Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary ( a closed 3-manifold ).

* William Jaco JSJ Decomposition of 3-manifolds This lecture gives a brief introduction to Seifert fibered 3-manifolds and provides the existence and uniqueness theorem of Jaco, Shalen, and Johannson for the JSJ decomposition of a 3-manifold.

Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric ; geometrically, it has one of 3 possible geometries: positive curvature / spherical, zero curvature / flat, negative curvature / hyperbolic – and the geometrization conjecture ( now theorem ) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.

Steenrod's theorem states that an orientable 3-manifold has a trivial tangent bundle.

This theorem is due independently to several people: it follows from the Dehn – Lickorish theorem via a Heegaard splitting of the 3-manifold.

According to the Lickorish – Wallace theorem any closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere.

In mathematics, the Lickorish – Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ± 1 surgery coefficients.

A corollary of the theorem is that every closed, orientable 3-manifold bounds a simply-connected compact 4-manifold.

This conjecture can be thought of as a first step to resolving the Poincaré conjecture, since the Lickorish – Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link.