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 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.

* Hausdorff maximal principle, a concept in mathematics

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.

In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal ( with respect to set inclusion ) amongst all proper ideals.

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element.

* Max (), an abbreviation for either maximal element, greatest element, or extreme value in mathematics ; see Maxima and minima

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.

In mathematics, under various anti-large cardinal assumptions, one can prove the existence of the canonical inner model, called the Core Model, that is, in a sense, maximal and approximates the structure of V. A covering lemma asserts that under the particular anti-large cardinal assumption, the Core Model exists and is maximal in a way.

In mathematics, the Frattini subgroup Φ ( G ) of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group e or the Prüfer group, it is defined by Φ ( G ) = G. It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of " small elements " ( see the " non-generator " characterization below ).

In mathematics, Frobenius ' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations.

Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity.

In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.

In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups.

In mathematics, the Kummer – Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number h < sub > K </ sub > of the maximal real subfield of the p-th cyclotomic field.

In mathematics, the Heawood number of a surface is a certain upper bound for the maximal number of colors needed to color any graph embedded in the surface.

In mathematics, the star product of two graded posets and, where has a unique maximal element and has a unique minimal element, is a poset on the set.

In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges.

Pythagoras believed that behind the appearance of things, there was the permanent principle of mathematics, and that the forms were based on a transcendental mathematical relation.

Despite these facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics.

The anthropic principle is often criticized for lacking falsifiability and therefore critics of the anthropic principle may point out that the anthropic principle is a non-scientific concept, even though the weak anthropic principle, " conditions that are observed in the universe must allow the observer to exist ", is " easy " to support in mathematics and philosophy, i. e. it is a tautology or truism

The originality of Descartes ' thinking, therefore, is not so much in expressing the cogito — a feat accomplished by other predecessors, as we shall see — but on using the cogito as demonstrating the most fundamental epistemological principle, that science and mathematics are justified by relying on clarity, distinctiveness, and self-evidence.

* Large deviation principle, the rate function in mathematics

The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

In mathematics, the principle that says that if the number of players is one more than the number of chairs, then one player is left standing, is the pigeonhole principle.

More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle.

Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism.

The Church – Turing thesis states that this is a law of mathematics — that a universal Turing machine can, in principle, perform any calculation that any other programmable computer can.

In mathematics, the Banach fixed-point theorem ( also known as the contraction mapping theorem or contraction mapping principle ) is an important tool in the theory of metric spaces ; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

This means that man becomes immortal only if and to the extent that he acquires knowledge of what he can in principle know, e. g. mathematics and the natural sciences.

According to Lewis ( 1918 ), the " principle of these diagrams is that classes set ( mathematics ) | set s be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.

Both Spinoza and Leibniz asserted that, in principle, all knowledge, including scientific knowledge, could be gained through the use of reason alone, though they both observed that this was not possible in practice for human beings except in specific areas such as mathematics.

Although the concepts of a law or principle in nature is borderline to philosophy, and presents the depth to which mathematics can describe nature, scientific laws are considered from a scientific perspective and follow the scientific method ; they " serve their purpose " rather than " questioning reality " ( philosophical ) or " statements of logical absolution " ( mathematical ).

Calculus, though allowed in solutions, is never required, as there is a principle at play that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge.

In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input ; a linear system fulfills these conditions.

* The superposition principle in physics, mathematics, and engineering, describes the overlapping of waves.

In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a smallest element.