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Closure (topology)

In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point, in the closure of S is a point of closure of S; the notion of closure is in many ways dual to the notion of interior. For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S; this definition generalizes to any subset S of a metric space X. Expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d < r. Another way to express this is to say that x is a point of closure of S if the distance d:= inf = 0; this definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood".

Let S be a subset of a topological space X. X is a point of closure of S if every neighbourhood of x contains a point of S. Note that this definition does not depend upon whether neighbourhoods are required to be open; the definition of a point of closure is related to the definition of a limit point. The difference between the two definitions is subtle but important — namely, in the definition of limit point, every neighbourhood of the point x in question must contain a point of the set other than x itself. Thus, every limit point is a point of closure. A point of closure, not a limit point is an isolated point. In other words, a point x is an isolated point of S if it is an element of S and if there is a neighbourhood of x which contains no other points of S other than x itself. For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S; the closure of a set S is the set of all points of closure of S, that is, the set S together with all of its limit points.

The closure of S is denoted cl, Cl, S ¯ or S −. The closure of a set has the following properties. Cl is a closed superset of S. cl is the intersection of all closed sets containing S. cl is the smallest closed set containing S. cl is the union of S and its boundary ∂. Set S is closed if and only. If S is a subset of T cl is a subset of cl. If A is a closed set A contains S if and only if A contains cl. Sometimes the second or third property above is taken as the definition of the topological closure, which still make sense when applied to other types of closures. In a first-countable space, cl is the set of all limits of all convergent sequences of points in S. For a general topological space, this statement remains true if one replaces "sequence" by "net" or "filter". Note that these properties are satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", "open". For more on this matter, see closure operator below.

Consider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere, it is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, the closed 3-ball - the closure of the 3-ball. The closure of the open 3-ball is the open 3-ball plus the surface. In topological space: In any space, ∅ = c l. In any space X, X = cl. Giving R and C the standard topology: If X is the Euclidean space R of real numbers cl =. If X is the Euclidean space R the closure of the set Q of rational numbers is the whole space R. We say that Q is dense in R. If X is the complex plane C = R2 cl =. If S is a finite subset of a Euclidean space cl = S. On the set of real numbers one can put other topologies rather than the standard one. If X = R, where R has the lower limit topology cl = [0, 1). If one considers on R the discrete topology in which every set is closed cl =. If one considers on R the trivial topology in which the only closed sets are the empty set and R itself cl = R.

These examples show. The last two examples are special cases of the following. In any discrete space, since every set is closed, every set is equal to its closure. In any indiscrete space X, since the only closed sets are the empty set and X itself, we have that the closure of the empty set is the empty set, for every non-empty subset A of X, cl = X. In other words, every non-empty subset of an indiscrete space is dense; the closure of a set depends upon in which space we are taking the closure. For example, if X is the set of rational numbers, with the usual relative topology induced by the Euclidean space R, if S = S is closed in Q, the closure of S in Q is S.

Gustav von Ewers

Johann Philipp Gustav von Ewers or Evers was a German legal historian and the founder of Russian legal history as a scholarly discipline. Ewers was a farmer’s son from the village of Amelunxen in the Bishopric of Paderborn, he first studied theology and political science at the University of Göttingen. His first employment, as was customary for a graduate from a poor background, was as a private tutor; this brought him to the Imperial Russian province of Livonia, where he was to remain for the rest of his life. While teaching, he pursued his scholarly interests regarding Russian political and legal history, which became one of his main fields of study – one indeed of which he is regarded the founder. Influenced by the Hegelian definition of society and state, he described the traditional tribal structure of Russia as the foundation of Russian statehood, most notably in this 1826 monograph Das älteste Recht der Russen. Evers' ideas have found a continued reception among Russian legal theorists.

On the basis of his publications, he was offered in 1810 the Chair of History and Geography of the Russian State at the University of Dorpat in what is today Estonia. He occupied that chair until 1826. In 1816, Ewers declined an offer of the Chair of Political Economy at the newly founded University of Berlin. In the same year, he had become Prorector of the University of Dorpat and in 1818, Rector, to which office he was re-elected every year until his death at Dorpat in 1830, aged 51. Provisorische Verfassung des Bauernstandes in Estland. 1805, 1806. Vom Ursprung des Russischen Staats. 1808 Unangenehme Erinnerungen an August Ludwig Schlözer. 1810. Kritische Vorarbeiten zur 2 vols. 1814. Geschichte der Russen, vol. 1. 1816. Das älteste Recht der Russen in seiner geschichtlichen Entwicklung. 1826. Rhapsodische Gedanken über die wissenschaftliche Bedeutung des Naturrechts. 1828. Djakonov, M. A.. "Johann Philipp Gustav v. Ewers." In G. V. Levitski, ed. Biografitsheskii Slovar professorov i prepodavatelei imperatorskavo Juerjevskago, byvshago Derptskago Univesriteta sa sto let ego sushestvovania, vol.

2. Jurjev: Mattisen, 510-537. Drechsler, Wolfgang. On the Eminence of the Social Sciences at the University of Dorpat. Aula Lectures. Tartu: at the University Press. Leppik, Lea. Rektor Ewers. Tartu: Eesti Ajalooarhiiv. Schevcov, Vladimir. Die Sippentheorie bei Gustav Ewers. Berlin. Stupperich, Robert. Gustav Ewers. Paderborn: Westfälische Lebensbilder. Peeter Järvelaid. Põhikooli õpetaja Gustav Ewers Väimelast. – Eesti Elu, 18 January 2008. Peeter Järvelaid. Õpetaja Gustav Ewers Väimelast. – Võrumaa Teataja, 12 January 2008. Meduschewski, Andrej. "Evers, Johann Gustav". In Michael Stolleis. Juristen: ein biographisches Lexikon. Jahrhundert. München: Beck. P. 100. ISBN 3-406-45957-9. Ostdeutsche Biographie EEVA

Mathilde Madden

Mathilde Madden, pen name of Mathilda Gregory, is a British erotica author and Guardian journalist. Her novel Equal Opportunities is about a female sadist who sexually dominates a man left wheelchair bound after an accident, her other novels include Mad About the Boy, about women paying men for sex and Peep Show, about a female voyeur who likes to spy on gay men. On writing erotica, Madden says, ``, you end up with your own tailor-made fun. My own work, because I’ve written it, touches every one of my hot buttons." She lives in Brighton and has two children. Her novel The Silver Crown was subjected to legal scrutiny when Pelican Bay State Prison, California withheld a copy from an inmate convicted of attempted murder, Andres Martinez, on grounds that the book was "obscene and to incite violence" and contained "descriptions of explicit sexual conduct." In May 2013, an appeals court affirmed the prisoner's right to read the book. Paranormal urban fantasy erotica novels: The Silver Collar The Silver Crown The Silver Cage Erotica novels: Peep Show Mad About the Boy Equal Opportunities Audio Plays: Doctor Who audio play Wet Walls for Short Trips - Volume 3 Official site

CIA activities in Canada

It has been traditionally believed that any U. S. Central Intelligence Agency activity in Canada would be undertaken with the "general consent" of the Canadian government, through the 1950s information was given to the CIA in return for information from the United States. However, traditionally Canada has refused to voice any anger when it was clear that the CIA was operating without any authorisation. Apologists have noted that Canada was vital to CIA operations as it "physically occupied the territory between the United States and the Soviet Union. However, on May 28, 1975 Solicitor General Warren Allmand directed the Royal Canadian Mounted Police to begin investigating the levels of CIA involvement in Canadian affairs. Today, the country continues to cooperate with the CIA, enabling their black sites, allowing ghost planes to land and refuel in Canada, en route to delivering prisoners to unknown locations; the Canadian counterpart of the CIA is the Canadian Security Intelligence Service and its agency cooperates with the CIA.

The CIA convinced the Allan Memorial Institute to allow a series of mind control tests on nine patients in the Montreal school, as part of their ongoing Project MKULTRA. The experiments were exported to Canada when the CIA recruited Scottish psychiatrist Donald Ewen Cameron, creator of the "psychic driving" concept, which the CIA found interesting. Cameron had been hoping to correct schizophrenia by erasing existing memories and reprogramming the psyche, he commuted from Albany, New York to Montreal every week and was paid $69,000 from 1957 to 1964 to carry out MKULTRA experiments there. In addition to LSD, Cameron experimented with various paralytic drugs as well as electroconvulsive therapy at thirty to forty times the normal power, his "driving" experiments consisted of putting subjects into drug-induced coma for weeks at a time while playing tape loops of noise or simple repetitive statements. His experiments were carried out on patients who had entered the institute for minor problems such as anxiety disorders and postpartum depression, many of whom suffered permanently from his actions.

His treatments resulted in victims' incontinence, forgetting how to talk, forgetting their parents, thinking their interrogators were their parents. When lawsuits commenced in 1986, the Canadian government denied having any knowledge that Cameron was being sponsored by the CIA; when the Avro Arrow aerospace program was cancelled in 1959, many believed that the CIA was responsible, fearing Canadian intrusion into aerospace dominance. In 1961, the CIA wrote an intelligence estimate titled "Trends in Canadian Foreign Policy" which suggested that the Conservative government of John Diefenbaker "might take Canada in a divergent direction" and seek "a more independent foreign policy" and suggested that a return of the Liberal Party might "soften the Canadian resistance to the storage of nuclear weapons on Canadian soil". In 1967, Prime Minister Lester Pearson announced he would investigate the allegations of the CIA helping him oust Diefenbaker. In 1982, Canadian Member of Parliament Svend Robinson accused the CIA of infiltrating the RCMP and funnelling political contributions to favoured politicians in provincial elections from 1970-76.

The information seemed to arise from John H. Meier, an aide to Howard Hughes, but a secret investigation turned up no evidence of such a conspiracy; the RCMP allegations dated back to 1977, when it was shown that they were "linked" to the CIA. By 1964, the CIA closely monitored the Canadian wheat industry, as the United States hoped to sell wheat to the Soviet bloc countries When the American embassy was seized by Iranian students in 1979, Canadian diplomat Kenneth D. Taylor was made the "de facto CIA Station Chief" in the country, but kept his new position secret from Canadians. An indication of the United States' close operational cooperation with Canada is the creation of a new message distribution label within the main US military communications network; the marking of NOFORN required the originator to specify which, if any, non-US countries could receive the information. A new handling caveat, USA/AUS/CAN/GBR/NZL eyes only, used on intelligence messages, gives an easier way to indicate that the material can be shared with Australia, Great Britain, New Zealand.

Aware that the Canadian Khadr family knew valuable intelligence about the inner workings of al Qaeda, the CIA hired Abdurahman Khadr to act as an informant and infiltrate Islamist circles. The CIA paid the Pakistani government $500,000 to capture and interrogate his older brother, Abdullah Khadr, ostensibly torturing him to secure answers and confessions; as of 2006, Canada had allowed 76 CIA flights to use the country's airbases in Nunavut and Labrador, to carry prisoners from the War on Terror to black sites overseas

Hey Deanie

"Hey Deanie" is a song written by Eric Carmen. It was a popular hit single by Shaun Cassidy, released the last week of November, 1977 from his album, Born Late, it was his third and final top 10 hit, peaked at number seven for two weeks on the Billboard Hot 100, spending four months on the chart from late 1977 to early 1978. As with Cassidy's prior singles, this song became a gold record."Hey Deanie" was the second hit written by Carmen for Cassidy, the first being "That's Rock'n' Roll," from Cassidy's previous album. Both songs charted concurrently with Carmen's own hit, "She Did It". Carmen recorded "Hey Deanie" himself for his 1978 album, Change of Heart, it was featured as the B side of the title track's 45 RPM release, which became a Top 20 hit. Carmen wrote the song after seeing the film, Splendor in the Grass, starring Natalie Wood as "Deanie" and Warren Beatty as her lover; as with Cassidy's bigger hits, his version of "Hey Deanie" was featured in one of the episodes of The Hardy Boys, as performed by Cassidy's character Joe Hardy.

The episode is entitled "Oh Say Can You Sing"."Hey Deanie" is ranked as the 68th biggest American hit of 1978. Chicago radio superstation WLS, which gave the song much airplay, ranked "Hey Deanie" as the 45th most popular hit of 1978, it reached as high as number 6 on their survey of January 21, 1978. While not charting nationally in Australia, "Hey Deanie" did reach #21 on station 2NUR in Sydney, Australia. "Hey Deanie" was covered by power pop artist Gary Charlson as a live version. It was included on his Real Live album. Lyrics of this song at MetroLyrics Listen to "Hey Deanie" on YouTube Listen to "Hey Deanie" on YouTube "Hey Deanie" at Song Facts

Bislett Games

The Bislett Games is an annual track and field meeting at the Bislett Stadium in Oslo, Norway. One of the IAAF Golden League events, it is now part of the IAAF Diamond League, it is sponsored by ExxonMobil and known as the ExxonMobil Bislett Games. The first international athletics meeting at Bislett was held in 1924; until 1937 the competitions are known as "The American Meetings". Different organizers staged the meetings between 1947 and 1965 until the three athletics associations BUL, Vidar and Tjalve formed the Bislett Alliance. At this year Arne Haukvik founded the Bislett Games, he was a former politician and director of the meeting, who used to invite the athletes and the press to his home for his traditional "strawberry party" the day before the event each year. He died of cancer in 2002 at age 76; the tradition however is continued. Bislett Stadium was used for speed skating events at the Olympics, but nowadays it is better known for its Bislett Games athletics meeting. Bislett Games attract the best track and field athletes from all over the world, 65 world records have been set on its forgiving, brick-coloured track so far.

Due to the building of the new Bislett Stadium in Oslo, which started in April 2004, the 2004 edition of the traditional athletics meeting was staged on Fana stadion in Bergen under the name Bergen Bislett Games. In 2009, a severe storm delayed proceedings and caused damage to the track-side clock display. Sanya Richards recorded the fastest women's 400 metres time since 2006 while the Dream Mile brought a number of records with winner Deresse Mekonnen improving upon his Ethiopian record, Kenyan William Biwott Tanui setting a world junior record and third-placed Augustine Choge beating his personal best. Former javelin winners Andreas Thorkildsen and Tero Pitkämäki continued their five-year shared dominance of the Bislett Games, with Pitkämäki taking the victory this time. Over the course of its history, numerous world records have been set at the Games and former athletics meetings at Bislett stadium. 1985 three new records was set at the same evening. + = en route to a longer distance Dream Mile Diamond League – Oslo Official Web Site