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Tychonoff's theorem

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case; the earliest known published proof is contained in a 1937 paper of Eduard Čech. Several texts identify Tychonoff's theorem as the single most important result in general topology; the theorem depends crucially upon the precise definitions of compactness and of the product topology. Conversely, part of its importance is to give confidence that these particular definitions are the most useful ones. Indeed, the Heine–Borel definition of compactness—that every covering of a space by open sets admits a finite subcovering—is recent. More popular in the 19th and early 20th centuries was the Bolzano–Weierstrass criterion that every sequence admits a convergent subsequence, now called sequential compactness.

These conditions are equivalent for metrizable spaces, but neither one implies the other in the class of all topological spaces. It is trivial to prove that the product of two sequentially compact spaces is sequentially compact—one passes to a subsequence for the first component and a subsubsequence for the second component. An only more elaborate "diagonalization" argument establishes the sequential compactness of a countable product of sequentially compact spaces. However, the product of continuum many copies of the closed unit interval fails to be sequentially compact with respect to the product topology though it is compact by Tychonoff's theorem; this is a critical failure: if X is a regular Hausdorff space, there is a natural embedding from X into C, where C is the set of continuous maps from X to. The compactness of C thus shows that every regular Hausdorff space embeds in a compact Hausdorff space This construction is the Stone–Čech compactification. Conversely, all subspaces of compact Hausdorff spaces are regular Hausdorff, so this characterizes the regular Hausdorff spaces as those that can be compactified.

Such spaces are now called Tychonoff spaces. Tychonoff's theorem has been used to prove many other mathematical theorems; these include theorems about compactness of certain spaces such as the Banach–Alaoglu theorem on the weak-* compactness of the unit ball of the dual space of a normed vector space, the Arzelà–Ascoli theorem characterizing the sequences of functions in which every subsequence has a uniformly convergent subsequence. They include statements less related to compactness, such as the De Bruijn–Erdős theorem stating that every minimal k-chromatic graph is finite, the Curtis–Hedlund–Lyndon theorem providing a topological characterization of cellular automata; as a rule of thumb, any sort of construction that takes as input a general object and outputs a compact space is 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, the Berkovich spectrum of a commutative Banach ring. 1) Tychonoff's 1930 proof used the concept of a complete accumulation point.

2) The theorem is a quick corollary of the Alexander subbase theorem. More modern proofs have been motivated by the following considerations: the approach to compactness via convergence of subsequences leads to a simple and transparent proof in the case of countable index sets. However, the approach to convergence in a topological space using sequences is sufficient when the space satisfies the first axiom of countability, but not otherwise. However, the product of uncountably many metrizable spaces, each with at least two points, fails to be first countable. So it is natural to hope that a suitable notion of convergence in arbitrary spaces will lead to a compactness criterion generalizing sequential compactness in metrizable spaces that will be as applied to deduce the compactness of products; this has turned out to be the case. 3) The theory of convergence via filters, due to Henri Cartan and developed by Bourbaki in 1937, leads to the following criterion: assuming the ultrafilter lemma, a space is compact if and only if each ultrafilter on the space converges.

With this in hand, the proof becomes easy: the image of an ultrafilter on the product space under any projection map is an ultrafilter on the factor space, which therefore converges, to at least one xi. One shows that the original ultrafilter converges to x =. In his textbook, Munkres gives a reworking of the Cartan–Bourbaki proof that does not explicitly use any filter-theoretic language or preliminaries. 4) Similarly, the Moore–Smith theory of convergence via nets, as supplemented by Kelley's notion of a universal net, leads to the criterion that a space is compact if and only if each universal net on the space converges. This criterion leads to a proof of Tychonoff's theorem, which is, word for word, identical to the Cartan/Bourbaki proof using filters, save for the repeated substitution of "universal net" for "ultrafilter base". 5) A proof using nets but not universal nets was given in 1992 by Paul Chernoff. All of the above proofs use the axiom of cho


Nestopia is an open-source NES/Famicom emulator designed to emulate the NES hardware as as possible. The requirements for Nestopia were considered higher than some of its contemporaries such as Stella. In order to run an optimal emulation, the program requires a minimum 800 MHz processor, its high requirement is due to its accurate emulation of the NES hardware. The emulator has a strong port for the Apple Macintosh. Nestopia allows customization of colors and graphics, it includes special features such as Power Glove. Brandon Widdler of Digital Trends considers the emulator one of the best for the NES, though he admits that it has fewer features than its rival FCEUX. Nestopia was developed for Windows by Martin Freij. Richard Bannister and R. Belmont ported it to Mac OS X and Linux, respectively. Original developers ended development in 2008, but a number of unofficial forks have been derived from the latest source. One unofficial continuation is being developed on Github. Nestopia has a RetroArch/Libretro port.

List of Nintendo Entertainment System emulators List of Nintendo Entertainment System games Official website

The Curiosity Show

The Curiosity Show is an Australian educational children's television show produced from 1972 to 1990 and hosted by Rob Morrison and Deane Hutton. The show was produced by Banksia Productions in South Australia for the Nine Network. 500 episodes were produced. Banksia Productions produced the popular children's series Here's Humphrey from 1965; the company planned to add some science segments in 1971 and sought assistance from the South Australian Institute of Technology. Rob Morrison and Deane Hutton were selected as presenters and the segments were introduced as Humphrey B Bear's Curiosity Show. After positive reception from the audience, Banksia Productions and the Nine Network agreed to produce a spin-off series. Planning commenced with the working title The F Show; until the early 1970s, children's television was aimed at younger children. The broadcasting regulations were changed to require a proportion of programmes to be aimed at school-age children, broadcast after school hours; this prompted the creation of the Curiosity Show as a separate show.

From 1972 to 1980 the format was a 60-minute show presented by Morrison, Ian Fairweather, Alister Smart, Belinda Davey, Gabrielle Kelly, Dr Mark Dwyer and Lynn Weston. The emphasis was on science but included general craft and music. Producers were Kate Kennedy White, James Lingwood and Ian Smyth. From 1980 the show was reduced to 30 minutes, presented by Morrison and Hutton, with emphasis on science and the environment; the Curiosity Show won many national and international awards, including the coveted Prix Jeunesse in 1984, voted by peers from around the world as the best factual program for children. The program placed a strong emphasis on practical demonstrations of various science topics, included activities such as floating a ping-pong ball on a stream of air, recreating historical devices, setting off a room full of mouse traps, the science of musical instruments and freezing objects with liquid nitrogen. Segments presented scientific concepts in the form of tricks and puzzles. Many segments described a sequence of steps to build something out of common household materials with longer builds invariably ending with the phrase "here's one I prepared earlier" so as to keep the segment moving.

Hutton's catchphrase, after presenting a hypothesis he postulates the audience may be curious about, was to declare "well, I'm glad you asked," before responding to the hypothesis. The use of household materials was deliberate, in order to demystify science and ensure that children, wherever they lived, could make what they needed rather than rely on buying it, this proved popular with the young audience who could replicate the demonstrations at home. Morrison suggested that they should always show what they had made working so that children would know that their own constructions would work if made properly and to show the limitations of the constructions to dispel any overambitious expectations. Both Morrison and Hutton always told viewers to get their parents' permission before building things or conducting experiments if it involved the use of sharp objects such as knives or scissors or the use of flames or hot or dangerous liquids. Four companion books were available in 1981 produced by Jacaranda Press containing scientific explanations and instructions for experiments for children to perform at home.

Each of the books was themed upon one of the four Western classical elements of earth, air and water. Together and Morrison published 11 books, including Supermindstretchers, The Arrow book of Things to Make and Do and String for Lunch. Morrison published more than 40 additional books which included material from Curiosity Show, including Nature in the Making, A Field Guide to the Tracks and Traces of Australian Animals the first such field guide in Australia and still the only one to deal with all taxa and Quirky Creatures, It's Raining Frogs and Fishes and many more for the school reading programs of various publishers. Morrison's field guide to Tracks and Traces, a segment he had prepared for Curiosity Show on dingos at Uluru, led to his involvement in the Morling enquiry into the Chamberlain Convictions, he gave evidence on dingo and dog tracks and conducted various forensic investigations on dingo gapes and behaviour. His collection of specimens and artefacts from the trial have been acquired by the National Museum of Australia.

The Children's Television Workshop wanted to make a version of the Curiosity Show using American scientists as presenters. Rob Morrison and Deane Hutton were consultants in the early planning stages. PBS didn't think that middle-aged scientists would engage a young audience and insisted that any science show be hosted/presented by young people. CTW reworked the concept into 3-2-1 Contact. In 2014, The Curiosity Show made a brief online series, available on YouTube, it was co-hosted by Hutton again. In 2013, the show's former hosts and Morrison, announced they had purchased the remaining rights to the show for an undisclosed sum, from Banksia Productions which had gone into liquidation. On July 12, 2013, in conjunction with producer Enabled Solutions, they launched a YouTube channel to make the episodes and segments available for a new generation of viewers. There are some 5,000 segments, some have attracted significant audiences in the United States and India. In May 2014, Hutton and Morrison released, on the YouTube channel, the "Curiosity Show's first new episode since 1990".

The online production was funded with the assistance of Kellogg Australia. The episode follow

Joaquín Pérez Martín Parapar

Joaquín Pérez Martín-Parapar was a Spanish railway and military man. Railroad by profession, 1 after the outbreak of the Civil War joined the Republican militias, he came to join the PCE. In December 1936 he was appointed commander of the newly created 66th Mixed Brigade.2 He led this unit in the battle of Jarama, in heavy fighting in the area of Esplegares. According to Carlos Engel, he would have commanded the 56th Mixed Brigade, reserve unit of the Army of Extremadura.3 In the spring of 1938 he was appointed commander of the 63rd Mixed Brigade, acting on the Levante front and in the Extremadura front. On April 5, its 250th Battalion attacked Carrascalejo prematurely, which it managed to conquer, but lost it on the 8th due to a nationalist counterattack; as a result of this operation, his boss, Major Blas, was dismissed, replaced by Joaquín Pérez Martín-Parapar, the eldest of the militias. The 56th Mixed Brigade was formed in January 1937, in the sub-sector of Almería, under the command of the Joaquín Pérez Martín de Parapar major.

He led the 66 Mixed Brigade in the Jarama. In December 1937, it became part of the reserve of the Army of Extremadura and was dissolved in April 1938, changing the largest unit. At the beginning of May, he left for the Levante front, joining the 48th Division of the XVI Army Corps and heading to Segorbe. Since June 11 he was defending the roads of Villafamés until the front stabilized. Once stabilized the Levantine front with the beginning of the Battle of the Ebro, returned to Extremadura. From August 12 to 18, he intervened in an operation on the Alto del Buitre that he conquered and lost again. On December 15, he was appointed to cover the passive front of Hinojosa del Duque, in which he remained successful during the battle of Peñarroya and stayed there until the end of the war. Captured by the Francoists at the end of the war, as they always did to the one who had confronted them, he was decreed guilty of rebellion armed with the maximum penalty, the execution in the city of Guadalajara in early 1941.

Comín Colomer, 1967, p. 57 Engel, 1999, p. 67 Engel, 1999, p. 59 Engel, 1999, p. 65 Alpert, Michael. El Ejército republicano en la guerra civil. Siglo XXI Editores. Álvarez, Santiago. Los comisarios políticos en el Ejército Popular de la República. Ediciós do Castro. Engel, 1999, p. 59. Martínez Bande, 1978, p. 285. Engel, 1999, p. 59-60. Martínez Reverte, 2006, p. 53. Engel, 1999, p. 60. Martínez Bande, José Manuel. La Batalla del Ebro. Madrid: ED. San Martín. Martínez Reverte, Jorge. La caída de Cataluña. Barcelona: Crítica. Salas Larrazábal, Ramón. Historia del Ejército Popular de la República. La Esfera de los Libros. Zaragoza, Cristóbal. Ejército Popular y Militares de la República, 1936-1939. Barcelona: Ed. Planeta. Comín Colomer, Eduardo. History of the Communist Party of Spain III. Madrid: National Publisher. Engel, Carlos. History of the mixed brigades of the People's Army of the Republic. Madrid: Battlement. ISBN 84-922644-7-0

Kellogg Brothers

The Kellogg Brothers were a family of lithographers and printmakers in Hartford, Connecticut from about 1830 to the end of the 19th century. The brothers were Jarvis Griggs Kellogg, Daniel Wright Kellogg, Edmund Burke Kellogg, Elijah Chapman Kellogg, they operated in a series of partnerships, between themselves and with others, the firms having a variety of names that involved "Kellogg". They issued decorative prints. Finlay, Nancy, ed. 2009. Picturing Victorian America: Prints by the Kellogg Brothers of Hartford, Connecticut, 1830-1880. Hartford: The Connecticut Historical Society. Catlini, Robert. "History Out Of Prints: Kellogg Brothers Exhibit Shows Vast Scope Of Printmaker's Output And Influence". Hartford Courant. Retrieved 28 October 2015

Tristan und Isolde discography

This is an audio and video discography of Tristan und Isolde, an opera by Richard Wagner, first performed on 10 June 1865 in Munich. Tristan und Isolde has a long recorded history. In the years before World War II, Kirsten Flagstad and Lauritz Melchior were considered to be the prime interpreters of the lead roles, mono recordings exist of this pair in a number of live performances led by conductors such as Thomas Beecham, Fritz Reiner, Artur Bodanzky and Erich Leinsdorf. Flagstad recorded the part commercially only near the end of her career in 1952, under Wilhelm Furtwängler for EMI, producing a set, considered a classic recording. Following the war, the performances at Bayreuth with Martha Mödl and Ramon Vinay under Herbert von Karajan were regarded, these performances are now available as a live recording. In the 1960s, the soprano Birgit Nilsson was considered the major Isolde interpreter, she was partnered with the Tristan of Wolfgang Windgassen, their performance at Bayreuth in 1966 under the baton of Karl Böhm was captured by Deutsche Grammophon—a performance hailed as one of the best Tristan recordings.

Some collectors prefer the pairing of Nilsson with the Canadian tenor Jon Vickers, available in "unofficial" recordings from performances in Vienna or Orange. There are several DVD productions of the opera including Götz Friedrich's production at the Deutsche Oper in Berlin featuring the seasoned Wagnerians René Kollo and Dame Gwyneth Jones in the title roles. Deutsche Grammophon released a DVD of a Metropolitan Opera performance featuring Jane Eaglen and Ben Heppner, conducted by James Levine, in a production staged by Jürgen Rose, a DVD of the 1995 Bayreuth Festival production with conductor Daniel Barenboim and featuring Waltraud Meier as Isolde and Siegfried Jerusalem as Tristan, staged by Heiner Mueller. Conductor: Pierre Boulez. Soloists: Wolfgang Windgassen. Recorded, Osaka, 10 April 1967. Wieland Wagner directed the production. Conductor: Karl Böhm. Soloists: Jon Vickers. New Philharmonia Chorus. Théâtre Antique, France, 7 July 1973; this is a valued video recording due to its excellent performance despite some technical problems.

DVD: Hardy Classic Video HCD 40009 is a good print Conductor: Daniel Barenboim, Orchester der Bayreuther Festspiele and Directed by: Jean-Pierre Ponnelle, Soloists: René Kollo, Johanna Meier, Matti Salminen, Hermann Becht, Hanna Schwarz, Unitel 1983, Laserdisc Philips 070-509-1 Conductor: Jiří Kout, Orchestra & Chorus of the Deutsche Oper Berlin and Directed by: Götz Friedrich, Soloists: René Kollo, Dame Gwyneth Jones, Robert Lloyd, Gerd Feldhoff, Hanna Schwarz. TDK 1993 DVD Conductor: Zubin Mehta. Opus Arte DVD 1998. Conductor: Peter Schneider. Soloists: Robert Dean Smith, Iréne Theorin, Michelle Breedt, Robert Holl, Jukka Rasilainen. Staged and directed by Christoph Marthaler. 3-DVD set: Opus Arte, ASIN: B002QEXC6W 2009. Notes SourcesBrown, Jonathan. Tristan und Isolde on record: a comprehensive discography of Wagner's music drama with a critical introduction to the recordings. Westport, Conn.: Greenwood Press. ISBN 978-0-313-31489-6. Recordings of Tristan und Isolde, Jonathan Brown's discography