Further, there is a generalization of the Stone – Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone – Weierstrass theorem and described below.

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 ).

However, instead of simply considering the space of ultrafilters on, the right way to generalize this construction is to consider the Stone space of the measure algebra of: the spaces and are isomorphic as C *- algebras as long as satisfies a reasonable finiteness condition ( that any set of positive measure contains a subset of finite positive measure ).

The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications ; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone – Čech compactification which exists for any Tychonoff space, a much larger class of spaces.

The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem ( which says that the product of compact spaces is compact ) applied to compact Stone spaces ; hence, the theorem's name.

* Stone spaces

* The category of Boolean algebras and Boolean homomorphisms is equivalent to the opposite of the category of Stone spaces and continuous functions.

One obtains a duality between the category of Boolean algebras ( with their homomorphisms ) and Stone spaces ( with continuous mappings ).

Probably the most general duality which is classically referred to as " Stone duality " is the duality between the category Sob of sober spaces with continuous functions and the category SFrm of spatial frames with appropriate frame homomorphisms.

* When restricting further to coherent sober spaces which are Hausdorff, one obtains the category Stone of so-called Stone spaces.

This section motivates and explains one of the most basic constructions of Stone duality: the duality between topological spaces which are sober and frames ( i. e. complete Heyting algebras ) which are spatial.

* Stone duality, on the duality between topological spaces which are sober and frames ( i. e. complete Heyting algebras ) which are spatial.

For any Boolean algebra B, S ( B ) is a compact totally disconnected Hausdorff space ; such spaces are called Stone spaces ( also profinite spaces ).

Restating the theorem using the language of category theory ; the theorem states that there is a duality between the category of Boolean algebras and the category of Stone spaces.

This duality means that in addition to the isomorphisms between Boolean algebras and their Stone spaces, each homomorphism from a Boolean algebra A to a Boolean algebra B corresponds in a natural way to a continuous function from S ( B ) to S ( A ).

* The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem.

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

The Stone – Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval, an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C ( X ) is investigated.

The Stone – Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space.

Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C ( X, R ) of real-valued continuous functions on X, with the topology of uniform convergence.

A version of the Stone – Weierstrass theorem is also true when X is only locally compact.

In fact, the space of all Lipschitz functions on a compact metric space is dense in the Banach space of continuous functions, an elementary consequence of the Stone – Weierstrass theorem.

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

As a rule of thumb, any sort of construction that takes as input a fairly general object ( often of an algebraic, or topological-algebraic nature ) and outputs a compact space is likely to use Tychonoff: e. g., the Gelfand space of maximal ideals of a commutative C * algebra, the Stone space of maximal ideals of a Boolean algebra, and the Berkovich spectrum of a commutative Banach ring.

The Stone representation of a Boolean algebra is always separative and compact ; the corresponding Boolean space is known as the Stone space of the Boolean algebra.

Topological fields of sets that are separative, compact and algebraic are called Stone fields and provide a generalization of the Stone representation of Boolean algebras.

UCE has also challenged the validity of the tribal-state compact between the Oneidas and New York State that Turning Stone Casino & Resort operates under.

Rolling Stone gave Zaireeka four stars out of five, writing: " Zaireekas wall-of-surround-sound approach melds droning-rock dissonance with warped, off-kilter pop melodies, producing a totally immersing post-Pet Sounds audio séance.

Lester Bangs favourably reviewed the album in Rolling Stone, writing that it was " the kind of album that sometimes insinuates itself into your routine with a totally unexpected thrust of musical power.

* A Boolean algebra is complete if and only if its Stone space of prime ideals is extremally disconnected.

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

In case X is a completely regular Hausdorff space, the Stone – Čech compactification can be identified with the spectrum of C < sub > b </ sub >( X ).

In 1803, Barton W. Stone led a group of revivalist New Light Presbyterian ministers to form independent Springfield Presbytery which eventually became the Christian Church ( Disciples of Christ ).

Study of the Stone Age has never been mainly about stone tools and archaeology, which are only one form of evidence.

A form of the axiom of choice is required to prove that every topological space has a Stone – Čech compactification.

Vulcans subsequently practiced a form of polytheism ; this can be seen in gods of war, peace, and death depicted on the Stone of Gol, as well as the celebration of Rumarie.

His success as a practitioner of modern architecture and his prominence as an academic, enabled Stone to form bonds with other academics of the era like Walter Gropius ( Chairman of the Department of Architecture at Harvard University's Graduate School of Design ), Pietro Belluschi ( Dean of MIT's School of Architecture and Planning ), George Howe ( Chairman of Yale University ’ s School of Architecture ) and William Wurster ( co-founder of the University of California at Berkeley College of Environmental Design ).

Stone's legacy — of giving form to the aspirations of the emerging consumer culture and of reconciling Modernism with the dynamism of the age — is established in the critical survey, Edward Durell Stone: Modernism's Populist Architect by Mary Anne Hunting, PhD.

Jewish connection and veneration to the site arguably stems from the fact that it contains the Foundation Stone which, according to the rabbis of the Talmud, was the spot from where the world was created and expanded into its current form.

Former members of Ronstadt's backing band went on to form the Eagles ( made up of members of the Flying Burrito Brothers, Poco and Stone Canyon Band ), who emerged as one of the most successful rock acts of all time, producing albums that included Desperado ( 1973 ) and Hotel California ( 1976 ).

The huge nearby granite dome, Stone Mountain, is composed largely of a rock called Lithonia gneiss, a form of granite.

" Simon Reynolds commented in Rolling Stone that the group's earlier records form " an endlessly seductive body of work that sounds always the same, always different.

Portions of the township have been taken to form Anglesea Borough ( on June 13, 1885 ; now North Wildwood City ), Avalon Borough ( April 18, 1892 ), Wildwood ( May 1, 1895 ), Stone Harbor Borough ( April 3, 1914 ) and West Wildwood ( April 21, 1920 ).

Stone Harbor is governed under the Borough form of New Jersey municipal government.

Prior to the 2011 school year Appalachia High School was merged with Powell Valley High School in neighboring Big Stone Gap to form Union High School.

In its early stages, the Court held the view that interstate commerce was wholly immune from state taxation " in any form ," " even though the same amount of tax should be laid on ( intrastate ) commerce ," This position gave way in time to a less uncompromising but formal approach, according to which, for example, the Court would invalidate a state tax levied on gross receipts from interstate commerce, or upon the " freight carried " in interstate commerce, but would allow a tax merely measured by gross receipts from interstate commerce as long as the tax was formally imposed upon franchises, or "' in lieu of all taxes upon ( the taxpayer's ) property ,'" Dissenting from this formal approach in 1927, Justice Stone remarked that it was " too mechanical, too uncertain in its application, and too remote from actualities, to be of value.

Stone learned enough Hebrew and Greek to read passages of the Bible in an earlier form, and determined that the Bible was ' friendly to women '.

During the Civil War, Stone joined with Elizabeth Cady Stanton, Susan B. Anthony, Martha Coffin Wright, Amy Post, Antoinette Brown Blackwell, Ernestine Rose, and Angelina Grimké Weld to form the Woman's National Loyal League in 1863.

Once Reconstruction began, Stone helped form the American Equal Rights Association ( AERA ).

" Stone met with Carrie Chapman Catt and Abigail Scott Duniway to form a plan for organizing in Colorado, and Stone attended two days of meetings about getting a woman suffrage drive re-started in Kansas.

The probe is named after the Rosetta Stone, as it is hoped the mission will help form an idea of how the solar system looked before planets formed.

Evidence suggests the diet of Stone Age humans did include, in some form, the refined starches and grains that are excluded from the Paleolithic diet.

Julia Kenna reviewed the book for Rolling Stone, commenting, " Full of contradictions and witty one-liners, Penman uses language as an art form, playing with puns, synonyms, repetition, and punctuation for added effect ... Two decades of politics, music and pop culture with a whip-smart wit and wisdom that draws you in and doesn ’ t let go.

Pete Fender subsequently went on to form Rubella Ballet with Gem Stone ( bass ) and Sid Ation ( drums ) who was later also in Flux of Pink Indians.

Another variant was used by T. H. White in The Sword in the Stone, where Merlin and Madam Mim fought a wizards ' duel, in which the duelists would endlessly transform until one was in a form that could destroy the other.