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

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.

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.

The prototypical example of a Banach algebra is, the space of ( complex-valued ) continuous functions on a locally compact ( Hausdorff ) space that vanish at infinity.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

* Uniform algebra: A Banach algebra that is a subalgebra of C ( X ) with the supremum norm and that contains the constants and separates the points of X ( which must be a compact Hausdorff space ).

Equipped with the topology of pointwise convergence on A ( i. e., the topology induced by the weak -* topology of A < sup >∗</ sup >), the character space, Δ ( A ), is a Hausdorff compact space.

F is a constant and D is a parameter that Richardson found depended on the coastline approximated by L. He gave no theoretical explanation but Mandelbrot identified L with a non-integer form of the Hausdorff dimension, later the fractal dimension.

* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

* The structure space of a commutative unital Banach algebra is a compact Hausdorff space.

By the same construction, every locally compact Hausdorff space X is a dense subspace of a compact Hausdorff space having at most one point more than X ..

* A compact subset of a Hausdorff space is closed.

This is to say, compact Hausdorff space is normal.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

) In particular, every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.

* A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.

A Hausdorff space is H-closed if every open cover has a finite subfamily whose union is dense.

Since every compact Hausdorff space is a Tychonoff space, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space.

Hemingway's fiction is supported by a `` moral '' backbone and in its search for ultimate meaning hints at a religious dimension.

Besides the lack of an adequate ethical dimension to the Governor's case, one can ask seriously whether our lead over the Russians in quality and quantity of nuclear weapons is so slight as to make the tests absolutely necessary.

The inventory of tones is much smaller, and commonly the contrasts range along one single dimension, pitch level.

A `` mental image '' subconsciously impressing us from beneath its language symbols in wakeful thought, or consciously in light sleep, is actually not an image at all but is comprised of realities, viewed not in the concurrent sensory stream, but within the depths of the fourth dimension.

If the fourth dimension is a physical concept and not purely metaphysical, through what medium does it extend??

but when the sounds of the two speakers are allowed to mix, there is excellent depth and dimension to the music.

What was lacking was a real sense of phrase, the kind of legato singing that would have added a dimension of smoothness to what is, after all, a very oily character.

against multiple comparisons in one dimension, it is not conservative

In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension.

In C ++ ( 1983 ), class templates exist for multi-dimensional arrays whose dimension is fixed at runtime as well as for runtime-flexible arrays.

The number of indices needed to specify an element is called the dimension, dimensionality, or rank of the array.

A one-dimensional array ( or single dimension array ) is a type of linear array.

The addressing formula is completely defined by the dimension d, the base address B, and the increments c < sub > 1 </ sub >, c < sub > 2 </ sub >, …, c < sub > k </ sub >.

This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.

Being to Hobbes ( and the other empiricists ) is the physical universe: The world, ( I mean ... the Universe, that is, the whole masse of all things that are ) is corporeall, that is to say, Body ; and hath the dimension of magnitude, namely, Length, Bredth and Depth: also every part of Body, is likewise Body ... and consequently every part of the Universe is Body, and that which is not Body, is no part of the Universe: and because the Universe is all, that which is no part of it is nothing ; and consequently no where.

In calculus, this picture also gives a geometric proof of the derivative if one sets and interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, where the coefficient of the linear term ( in ) is the area of the n faces, each of dimension

The n-dimensional generalization of the quadrant and octant is the orthant.

For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.

R < sup > n </ sup > is the prototypical example of a real n-dimensional vector space.

In fact, every real n-dimensional vector space V is isomorphic to R < sup > n </ sup >.

In general, when an n-dimensional grid network is connected circularly in more than one dimension, the resulting network topology is a torus, and the network is called " toroidal ".

* In an n-dimensional vector space the identity function is represented by the identity matrix I < sub > n </ sub >, regardless of the basis.

* The Euclidean group E < sub > n </ sub >( R ) is the Lie group of all Euclidean motions, i. e., isometric affine maps, of n-dimensional Euclidean space R < sup > n </ sup >.

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

In general, it is also called n-dimensional volume, n-volume, or simply volume.

A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R < sup > n </ sup >.

For example, suppose the state space is the n-dimensional complex Hilbert space C < sup > n </ sup > and A is a Hermitian matrix with eigenvalues λ < sub > i </ sub >, with corresponding eigenvectors ψ < sub > i </ sub >.

When referring to an n-dimensional generalization, the term n-polytope is used.

An n-dimensional polytope is bounded by a number of ( n − 1 )- dimensional facets.

Together, these first two examples give a different proof that n-dimensional Euclidean space is separable.

* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication

It turns out that this map is bijective and can thus be used to transfer the vector space operations from R < sup > n </ sup > over to T < sub > x </ sub > M, turning the latter into an n-dimensional real vector space.

Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices.