Historian Norman Cantor who specialized in the medieval period, teaching and writing at Columbia and New York University, says in 1993: " It may be true that the Arabs had now fully extended their resources and they would not have conquered France, but their defeat ( at Tours ) in 732 put a stop to their advance to the north.

`` It is as though '', I said on the historic three-hour, coast-to-coast radio broadcast which I bought ( following Father Coughlin and pre-empting the Eddie Cantor, Manhattan Merry-go-round and Major Bowes shows ) `` That Man in the White House, like some despot of yore, insisted on reading my diary, raiding my larder and ransacking my lingerie!!

In this sense almost all reals are not a member of the Cantor set even though the Cantor set is uncountable.

Cantor is quoted as saying:

Some believe that Georg Cantor's set theory was not actually implicated by these paradoxes ( see Frápolli 1991 ); one difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system.

It is undisputed that, by 1900, Cantor was aware of some of the paradoxes and did not believe that they discredited his theory.

However, the term naive set theory is also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory ; care is required to tell which sense is intended.

In mathematics, the continuum hypothesis ( abbreviated CH ) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets.

* The Cantor set is compact.

In fact, every compact metric space is a continuous image of the Cantor set.

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.

Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment.

Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments.

A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

The Cantor space is the collection of all infinite sequences of 0s and 1s.

The probability measure on Cantor space, sometimes called the fair-coin measure, is defined so that for any binary string x the set of sequences that begin with x has measure 2 < sup >-| x |</ sup >.

Cantor Fitzgerald L. P. is a global financial services firm specializing in bond trading.

The company's effort to regain its footing is the subject of Tom Barbash's 2003 book On Top of the World: Cantor Fitzgerald, Howard Lutnick, and 9 / 11: A Story of Loss and Renewal.

Cantor Gaming is a Cantor Fitzgerald affiliate that operates race and sports books.

( Konig is now remembered as having only pointed out that some sets cannot be well-ordered, in disagreement with Cantor.

Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica.

Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem.

Cantor developed important concepts in topology and their relation to cardinality.

His collection of American art was strong in Abstract expressionism, Pop Art, Minimalism, Neo-Dada, Color Field, Lyrical Abstraction, and Neo-Expressionism and he often donated important works from his collection to institutions like MoMA, and other important private museums and University collections like the Norton Simon Museum, the Sheldon Museum of Art and the Iris & B. Gerald Cantor Center for Visual Arts at Stanford University among many others.

An important example of a totally disconnected space is the Cantor set.

Two important members of the Notre Dame administration have been suggested as possible identities for Perotinus: the theologian Petrus Cantor ( who died in 1197 ) and the Petrus who was Succentor of Notre Dame from at least 1207 until about 1238.

Norman Cantor recognized three books as most important to Strayer's legacy: Feudalism ( 1965 ), which summarized three decades of his research and thinking on the topic ; On the Medieval Origins of the Modern State ( 1970 ), in which he shows the relevance of medieval historical institutions to modern governmental institutions ; and The Reign of Philip the Fair ( 1980 ), representing over 30 years of archival research and the most comprehensive work on the topic in any language-other than Jean Favier's Philippe le Bel ( 1978 ).

The choice of the Polish space in the third clause above is not very important ; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.

As with many other episodes in the series, " Thirty Minutes Over Tokyo " suggests that, in the end, the Simpsons are more attached to the local than to the global, and, as Cantor writes: " indeed the global is ultimately important in the series only insofar as it can be made local, that is, part of Springfield.

Bernoulli schemes are important in the study of dynamical systems, as most such systems ( such as Axiom A systems ) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift.

'' My impassioned plea for civil rights created a landslide of correspondence and one sponsor even asked me to consider replacing the Eddie Cantor comedy hour on a permanent basis.

This implies that for each natural number n, the set of sequences f in Cantor space such that f ( n )

On September 19, Cantor Fitzgerald made a pledge to distribute 25 percent of the firm's profits for the next five years, and committed to paying for ten years of health care, for the benefit of the families of its 658 former Cantor Fitzgerald, eSpeed, and TradeSpark employees ( profits which would otherwise have been distributed to the Cantor Fitzgerald partners ).

On September 2, 2004, Cantor filed a civil lawsuit against the Kingdom of Saudi Arabia for allegedly providing money to the hijackers and Al Qaeda, one of a number of organizations to do so.

In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.

Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a " corrupter of youth " for teaching his ideas to a younger generation of mathematicians.

Cantor came to believe that Kronecker's stance would make it impossible for Cantor ever to leave Halle.

Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly ( while Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare ), and this tragedy drained Cantor of much of his passion for mathematics.

Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria.

Before Cantor, there were only finite sets ( which are easy to understand ) and " the infinite " ( which was considered a topic for philosophical, rather than mathematical, discussion ).

By proving that there are ( infinitely ) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied.

In one of his earliest papers, Cantor proved that the set of real numbers is " more numerous " than the set of natural numbers ; this showed, for the first time, that there exist infinite sets of different sizes.