Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization.

Also important are Plemelj's contributions to the theory of analytic functions in solving the problem of uniformization of algebraic functions, contributions on formulation of the theorem of analytic extension of designs and treatises in algebra and in number theory.

The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces.

The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature.

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere.

The uniformization theorem implies a similar result for arbitrary connected second countable surfaces: they can be given Riemannian metrics of constant curvature.

Felix and conjectured the uniformization theorem for ( the Riemann surfaces of ) algebraic curves.

The first rigorous proofs of the general uniformization theorem were given by and.

However, his proof relied on the uniformization theorem.

that it is nevertheless possible to prove the uniformization theorem via Ricci flow.

Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere ( or equivalently if every Jordan curve separates it ), then it is conformally equivalent to an open subset of the complex sphere.

The simultaneous uniformization theorem of Bers shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus > 1 with the same quasi-Fuchsian group.

The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient.

) Indeed, a triumph of nineteenth century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane.

* uniformization theorem

When Yau was a graduate student, he started to generalize the uniformization theorem of Riemann surfaces to higher-dimensional complex Kähler manifolds.

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.

2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves ) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables ( complex surfaces ), though not every 4-manifold admits a complex structure.

In the context of compact Riemann surfaces X, via the Riemann uniformization theorem, this can be seen as a distinction between the surfaces of different topologies:

By the uniformization theorem, any hyperbolic surface X – i. e., the Gaussian curvature of X is equal to negative one at every point – is covered by the hyperbolic plane.

In mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic.

AD < sub > R </ sub > is equivalent to AD plus the axiom of uniformization.

In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if is a subset of, where and are Polish spaces,

Such a function is called a uniformizing function for, or a uniformization of.

A uniformization of then picks exactly one element from each such subset, whenever the subset is nonempty.

A pointclass is said to have the uniformization property if every relation in can be uniformized by a partial function in.

The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.

It follows from ZFC alone that and have the uniformization property.

Thus, allowing arbitrary sets X and Y ( rather than just Polish spaces ) would make the axiom of uniformization equivalent to AC.

Note that the term " uniformization " correctly suggests a kind of smoothing away of irregularities in the geometry, while the term " geometrization " correctly suggests placing a geometry on a smooth manifold.

In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.

The Ricci flow does not preserve volume, so to be more careful in applying the Ricci flow to uniformization and geometrization one needs to normalize the Ricci flow to obtain a flow which preserves volume.

* Fredkin's concept of the multiverse as a finite automaton with absolute space, time, and information might be isomorphic to a sheaf uniformization axiom.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

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 debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.