** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

# REDIRECT Hausdorff maximal principle

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 ( Moore 1982: 168 ).

The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over Zermelo – Fraenkel set theory.

The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is very similar to this proof.

es: Principio maximal de Hausdorff

Spec ( R ) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology.

* Hausdorff maximal principle

Under this duality, every compact Hausdorff space is associated with the algebra of continuous complex-valued functions on, and every commutative C *- algebra is associated with the space of its maximal ideals.

# REDIRECT Hausdorff maximal principle

The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma ( Kelley 1955: 33 ).

The concept was introduced in 1918 by the mathematician Felix Hausdorff.

* a mathematical concept used in topology, see Hausdorff space

* Hausdorff dimension, a measure theoretic concept of dimension

* Felix Hausdorff introduces the concept of the fractional Hausdorff dimension.

Thus, the concept of the Hausdorff measure generalizes counting, length, and area.

In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept is named after Harald Bohr who pioneered the study of almost periodic functions, on the real line.

* Separated space, a synonym for Hausdorff space, a concept in mathematical topology

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

In topology and related branches of mathematics, a Hausdorff space, separated space or T < sub > 2 </ sub > space is a topological space in which distinct points have disjoint neighbourhoods.

In mathematics, the Hausdorff dimension ( also known as the Hausdorff – Besicovitch dimension ) is an extended non-negative real number associated with any metric space.

In mathematics, the Baker – Campbell – Hausdorff formula is the solution to

In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π ( g ) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact ( Hausdorff ) topological group and the representations are strongly continuous.

In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu – Hausdorff distance, measures how far two subsets of a metric space are from each other.

In mathematics, Gromov – Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in R < sup > n </ sup > or, more generally, in any metric space.

In mathematics ( specifically, measure theory ), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.

In mathematics, the Hausdorff paradox, named after Felix Hausdorff, states that if you remove a certain countable subset of the sphere S < sup > 2 </ sup >, the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent.

In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large-scale behavior of the manifold resembles that of a Euclidean space ( unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space ).

However he later employed the mathematics of topological Hausdorff sets, interpreting them as a model for the value-structure of metaphor, in a paper on Aesthetics.

In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function has a fixed point.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** Antichain principle: Every partially ordered set has a maximal antichain.

The maximal set produced by the principle is not unique, in general ; there may be many maximal totally ordered subsets containing a given totally ordered subset.

An equivalent form of the principle is that in every partially ordered set there exists a maximal totally ordered subset.

The holographic principle was inspired by black hole thermodynamics, which implies that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected.

The principle states that systems found in the natural world can perform computations up to a maximal (" universal ") level of computational power.

The fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound ( wherein the Bekenstein bound becomes an equality ) was the main observation that led to the holographic principle.

The fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound ( wherein the Bekenstein bound becomes an equality ) was the main observation that led to the holographic principle.

These include Stein interpolation ( a variable-parameter version of complex interpolation ), the Stein maximal principle ( showing that under many circumstances, almost everywhere convergence is equivalent to the boundedness of

C syntax makes use of the maximal munch principle.

According to the principle of PECs, the two basic requirements for materials used as photo-electrodes are optical function, required to obtain maximal absorption of solar energy, and catalytic function, required for other reactions such as water decomposition.

* Initiative versus principle ; If Israel rejects the best Arab position, perhaps the Arabs should revert to maximal demands and ask Israel to propose a plan, 29 March 2007, Al-Ahram Weekly