Among those Hausdorff compactifications, there is a unique " most general " one, the Stone – Čech compactification βX.

It is characterised by the universal property that, given a continuous map f from X to any other compact Hausdorff space Y, there is a unique continuous map g from βX to Y that extends f in the sense that f is the composition of g and j.

If X is Tychonoff, then the uniform structure can be chosen so that βX becomes the completion of the uniform space X.

In the mathematical discipline of general topology, Stone – Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX.

The Stone – Čech compactification βX of a topological space X is the largest compact Hausdorff space " generated " by X, in the sense that any map from X to a compact Hausdorff space factors through

If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a ( dense ) subspace of βX.

As is usual for universal properties, this universal property, together with the fact that βX is a compact Hausdorff space containing X, characterizes βX up to homeomorphism.

* The map from X to its image in βX is a homeomorphism if and only if X is Tychonoff.

* The map from X to its image in βX is a homeomorphism to an open subspace if and only if X is locally compact Hausdorff.

The Stone – Čech construction can be performed for more general spaces X, but the map X → βX need not be a homeomorphism to the image of X ( and sometimes is not even injective ).

For general topological spaces X, the map from X to βX need not be injective.

The capacity for making the distinctions of which diplomacy is compact, and the facility with language which can render them into validity in the eyes of other men are the leader's means for transforming the moral intuition into moral leadership.

The purchase of compact ( economy ) cars is being made currently on a test basis.

They estimate further that with sufficient experience and when cost-data of compact cars is compiled, the break-even point may be reduced to 7,500 miles of travel per year.

Fury is upstanding and on the rangy side, and Caper is more the compact type.

Staley Hanover ( Knight Dream-Sweetmite Hanover ) is a little on the small side but a very compact colt and looks like one to stand training and many future battles with colts in his class.

These differences in turn result from the fact that my Yokuts vocabularies were built up of terms selected mainly to insure unambiguity of English meaning between illiterate informants and myself, within a compact and uniform territorial area, but that Hoijer's vocabulary is based on Swadesh's second glottochronological list which aims at eliminating all items which might be culturally or geographically determined.

In a way, we may be witnessing the same thing in the sales of automobiles today as the public no longer is willing to purchase any car coming on the market but is more insistent on compact cars free of the frills which were accepted in the Fifties.

It allows compact encoding, but is less reliable for data transmission ; an error in transmitting the shift code typically makes a long part of the transmission unreadable.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

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

** A uniform space is compact if and only if it is complete and totally bounded.

Its weight also makes it relatively easy to move and carry, however its shape is generally not very compact and it may be difficult to stow unless a collapsing model is used.

An even more compact representation is given by noticing that:

The Government of Burkina Faso is working closely with MCC staff to finalize its compact submission.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

** 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.