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Product topology

In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another more obvious, topology called the box topology, which can be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine. Given X such that X:= ∏ i ∈ I X i is the Cartesian product of the topological spaces Xi, indexed by i ∈ I, the canonical projections pi: X → Xi, the product topology on X is defined to be the coarsest topology for which all the projections pi are continuous; the product topology is sometimes called the Tychonoff topology. The open sets in the product topology are unions of sets of the form ∏ i ∈ I U i, where each Ui is open in Xi and Ui ≠ Xi for only finitely many i.

In particular, for a finite product, the set of all Cartesian products between one basis element from each Xi gives a basis for the product topology of ∏ i ∈ I X i. That is, for a finite product, the set of all ∏ i ∈ I U i, where U i is an element of the basis of X i, is a basis for the product topology of ∏ i ∈ I X i; the product topology on X is the topology generated by sets of the form pi−1, where i is in I and Ui is an open subset of Xi. In other words, the sets form a subbase for the topology on X. A subset of X is open if and only if it is a union of intersections of finitely many sets of the form pi−1; the pi−1 are sometimes called open cylinders, their intersections are cylinder sets. In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general, the box topology is finer than the product topology, but for finite products they coincide. If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn.

The Cantor set is homeomorphic to the product of countably many copies of the discrete space and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology. Several additional examples are given in the article on the initial topology; the product space X, together with the canonical projections, can be characterized by the following universal property: If Y is a topological space, for every i in I, fi: Y → Xi is a continuous map there exists one continuous map f: Y → X such that for each i in I the following diagram commutes: This shows that the product space is a product in the category of topological spaces. It follows from the above universal property that a map f: Y → X is continuous if and only if fi = pi ∘ f is continuous for all i in I. In many cases it is easier to check. Checking whether a map f: Y → X is continuous is more difficult. In addition to being continuous, the canonical projections pi: X → Xi are open maps.

This means. The converse is not true: if W is a subspace of the product space whose projections down to all the Xi are open W need not be open in X; the canonical projections are not closed maps. The product topology is called the topology of pointwise convergence because of the following fact: a sequence in X converges if and only if all its projections to the spaces Xi converge. In particular, if one considers the space X = RI of all real valued functions on I, convergence in the product topology is the same as pointwise convergence of functions. Any product of closed subsets of Xi is a closed set in X. An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces is compact; this is easy to show for finite products, while the general statement is equivalent to the axiom of choice. Separation Every product of T0 spaces is T0 Every product of T1 spaces is T1 Every product of Hausdorff spaces is Hausdorff Every product of regular spaces is regular Every product of Tychonoff spaces is Tychonoff A product of normal spaces need not be normal Compactness Every product of compact spaces is compact A product of locally com

Second Balkenende cabinet

The Second Balkenende cabinet was the cabinet of the Netherlands from 27 May 2003 until 7 July 2006. The cabinet was formed by the political parties Christian Democratic Appeal, People's Party for Freedom and Democracy and the Democrats 66 after the election of 2003; the centre-right cabinet was a majority government in the House of Representatives. On 24 January 2003 Queen Beatrix asked Minister of Justice Piet Hein Donner to lead the coalition negotiations; the negotiations for the coalition were lengthy. The CDA preferred to continue its Centre-right coalition with the VVD, but they did not have sufficient seats in the House of Representatives to continue in government without the support of a third party. Another coalition with Pim Fortuyn List would be to be unpopular with voters after the events of the First Balkenende cabinet, the D66 was unwilling to join such a coalition. A government supported by the Christian right-wing Reformed Political Party and the social conservative Christian Union was opposed by the VVD.

A long negotiation between CDA and the Labour Party followed. The CDA and PvdA had come out of the elections as equal partners; the negotiations were troubled by the invasion of Iraq, the bad economic forecasts and personal animosity between incumbent Prime Minister and Leader of the Christian Democratic Appeal Jan Peter Balkenende and Wouter Bos the Leader of the Labour Party. After a couple of months talks were called off by Balkenende. At this point, D66 decided to join the coalition after all; the cabinet was based on a slim majority in the House of Representatives of 78 seats out of 150. When VVD Member of the House of Representatives Geert Wilders left his party on 2 September 2004, the narrow majority of the cabinet slimmed down further to 77 seats in the Member of the House of Representatives; the cabinet program is based around the slogan: Meer Werk, Minder Regels. The cabinet seeks to address the problems of integration of ethnic minorities, the economic recession and the lack of trust in government.

The most controversial issue the cabinet addressed is the perceived lack of integration of ethnic minorities immigrants from Morocco and Turkey. To solve this problem this cabinet has tried to reduce the influx of migrants, to force migrants to take an integration course; the cabinet appointed Rita Verdonk as a Minister without Portfolio within the Ministry of Justice with the responsibilities for Integration and Asylum Affairs. The number of immigrants allowed into the Netherlands was reduced by enforcing the asylum-seekers law of 2000 rigidly; this law was created under the Second Kok cabinet by the Mayor of Amsterdam, Job Cohen. Controversially, 26,000 asylum-seekers who had lived in the Netherlands for over five years but who had not been granted asylum were deported. Furthermore, partners of Dutch citizens are only allowed to immigrate into the Netherlands if the Dutch partner earns more than 120 percent of the minimum income; this income requirement has been decreased back to 100% of the minimum wage in 2010 as a result of the judgement of the EU Court in the Chakroun case.

Since 2006, family migrants from'non-Western' countries who want to immigrate into the Netherlands must pass an integration test. It tests the applicant's knowledge of political system and social conventions; the test must be taken before entering the Netherlands, in a Dutch Embassy or Consulates in the country of origin. Once in the Netherlands, migrants were obliged to pass a second test before receiving a permanent residence permit or being allowed to naturalise.'Oudkomers', i.e. foreigners who have lived in the Netherlands for a long time, were obliged by the Dutch government to take the exam as well. The cabinet took power at a time when the Netherlands' economy was in poor shape, with increasing unemployment and slight economic contraction. In order to jump start economic growth, the cabinet has proposed tax cuts and reform of the system of social welfare; the cabinet has implemented a new law for disability pensions. Most people enjoyed disability pensions under the old disability law received pensions if they were only disabled and could still work.

The pensions of these people have been cut, so they are forced to return to the workforce. Furthermore, the cabinet has limited the possibility of early retirement. Without exception all Dutch employees will be forced to work until they have become 65 longer; the cabinet has cut government spending by 5700 million euro, making a total of 11 billion euro, when combined with the cuts announced by the previous cabinet. Among other measures, free dental care and anti-conception medication were cut, 12000 positions were to be eliminated in the armed forces and some of their bases closed, the link between benefit payment rates and salaries was to be broken, the rental housing subsidy was reduced. At the same time, 4 billion euro in extra spending was made available in education and justice. Another controversial issue is the reform of the Dutch political system; this was proposed in order to overcome the'gap between politics and citizens', which became clear in the 2002 elections, which were dominated by the populist Pim Fortuyn, Assassinated during the election campaign.

The cabinet appointed Thom de Graaf as Deputy Prime Minister and as a Minister without Portfolio within the Ministry of the Interior and Kingdom Relations with the responsibilities for Government Reform. Thom de Graaf. De Graaf, who proposed an ambitious reform proposal, was met with much resistance. Two of the mo

Kenaboi language

Kĕnaboi is an extinct unclassified language of Negeri Sembilan, Malaysia that may be a language isolate or an Austroasiatic language belonging to the Aslian branch. It is attested in what appears to be two dialects, based on word lists of about 250 lexical items collected around 1870–90. In Walter William Skeat and Charles Otto Blagden's 1906 work "Pagan Races of the Malay Peninsula", the contents of three unpublished wordlists appear, two of which were collected by D. F. A. Hervey, a former government official in Malacca. There is no indication as to. Hervey collected his Kenaboi lexicon in Alor Gajah, Melaka from speakers living in Gunung Dato', Rembau District, a small inland mountainous area in southern Negeri Sembilan. Based on the ethnonym, the Kenaboi may have originated from the Kenaboi River valley of Jelebu District, northern Negeri Sembilan. Today, the Orang Asli of Negeri Sembilan are Temuan speakers. Hajek proposes that Kenaboi is a mixed language of both Aslian and Austronesian origins, with Kenaboi having a higher proportion of Austroasiatic words than Kenaboi.

Kenaboi has many words of unknown origin, such as mambu'white' and par'water'. Hajek speculates that the lexical aberrancy of Kenaboi 1 may due to the fact that Kenaboi 1 was a special taboo language, while Kenaboi 2 was the regular non-taboo language; the lexicon of Kenaboi 1 is 47% Austroasiatic, 27% Austronesian, 26% unclassified out of a total of 216 words. Hammarström, et al. note in Glottolog that Kenaboi is best considered to be a language isolate, do not consider arguments of Kenaboi as a taboo-jargon to be convincing. Skeat and Blagden considers Kenaboi as an isolate unrelated to Austronesian. Rasa, another extinct language documented in Skeat & Blagden near Rasa in Ulu Selangor has many words of uncertain origin. Kenaboi word list Andamanese languages Philippine Negrito languages Proto-Aslian language Kusunda languageOther Southeast Asian languages with high proportions of unique vocabulary of possible isolate origin: Enggano language Manide language Umiray Dumaget language Benjamin, Geoffrey.

2006. Hervey's'Kenaboi': lost Malayan Language or forest-collecting Taboo Jargon?. Singapore. Hajek, John. 1996. The Mystery of the Kenaboi: A First Report. Royal Institute of Linguistics and Anthropology

Aiguille du Grépon

The Aiguille du Grépon, informally known as The Grepon, is a mountain in the Mont Blanc Massif in Haute-Savoie, France. The Grepon has a Southern and Northern peak, which are the highest points of a sharp granite ridge to the east of the Glacier des Nantillons above Chamonix and northeast of the Aiguille du Midi. A madonna statue is situated on the Southern peak; the first ascent was made in 1881 by the Swiss climbers Alexander Burgener and Benedikt Venetz guiding Albert F. Mummery from England; this team had climbed one of the peaks of the neighboring Aiguille des Grands Charmoz the previous year. Two days after an attempt on the East face was found too challenging, they climbed up the couloir separating the Charmoz and Grepon from the Nantillons site to climb the Grepon over the north ridge; the party took a difficult narrow chimney from just below the col between the peaks. Though Venetz discovered the route and was the lead climber through it, this vertical crevice became famous as Mummery's crack.

The party reached the Northern summit, built a cairn and returned, but Mummery wondered at night if the Southern summit they had seen may have been higher. Thus the party came back over the same route two days and completed a traverse to the highest crag, which involved some abseiling, "a broad road suitable for carriages, bicycles, or other similar conveyances", more spectacular final lead climbing by Venetz, who hauled Mummery to the top after he slipped halfway; the second ascent leader, François Henri Dunod brought up three ladders as he had heard stories of the difficulty of this last pitch, though it turned out there was an easier crack to the top. More than 20 years this rock climb was still considered to be one of the most difficult in the Alps. Other climbing events: 2 September 1885: After a month of persistent effort, François Henri Dunod, François et Gaspard Simond and Auguste Tairraz climb the main summit from the south west; this route is less difficult and is the normal route of descent.

Before 1887: An unknown party reached at least the Northern peak via the first ascent route, but avoiding the Mummery crack by driving wooden wedges into a crack on the East side of the ridge. 18 August 1892: Albert F. Mummery, John Norman Collie, Geoffrey Hastings and Henri Pasteur do the first North-South traverse. 4 August 1893: Lily Bristow is the first woman to climb the Grépon, traversing it with Mummery and William Cecil Slingsby, while taking pictures on the way 19 August 1911: First ascent East face by Alexis Brocherel and Josef Knubel guiding Humphrey Owen Jones, Ralph Todhunter and Geoffrey Winthrop Young. This route contains the "Knubel crack", the first V+ pitch in the Mont Blanc region. 16 April 1927: First winter ascent by Camille Devouassoux and Armand Charlet 25 July 1947: First ascent West face by Robert Gabriel, Georges Livanos, Roger Duchier and Charles Magol Since 1927, at the top of the needle stands a statue of Our Lady of La Salette, 1.20 m high. With its weight of 44 kg, it was mounted on the back of a man on June 22, 1927 by eight guides of Chamonix members of the Jeunesse catholique.

François Labande, La Chaîne Du Mont-Blanc, sélection de voies. Tome 1, A L'Ouest Du Col Du Géant. Arthaud Guide Vallot GHM 2005, ISBN 2700305671

Wartislaw VI, Duke of Pomerania

Wartislaw VI of Pomerania was a member of the House of Griffins. From 1365 to 1377, he ruled Pomerania-Wolgast jointly with his brother Bogislaw VI. From 1377 until his death, he was the sole ruler of Pomerania-Barth, he was the eldest son of the Duke Barnim IV of Pomerania-Wolgast-Rügen and his wife, Sophie of Werle. After the death of his father Barnim IV in 1365, Pomerania-Wolgast was divided in the 1372 Treaty of Anklam into the eastern Duchy of Pomerania-Stolp, ruled by his uncle Bogislaw V and the western Duchy of Pomerania-Wolgast, ruled jointly by Wartislaw VI and his younger brother Bogislaw VI. In 1377, Pomerania-Wolgast was divided into a smaller Pomerania-Wolgast, ruled by Bogislaw VI, Pomerania-Barth ruled by Wartislaw VI. In 1396, Bogislaw VI died at Klępino Białogardzkie, without a male heir, the two parts of Pomerania-Wolgast were reunited under Wartislaw VI, he married Anne of a daughter of Duke John I of Mecklenburg-Stargard. They had three children: Barnim VI, Duke of Pomerania Wartislaw VIII, Duke of Pomerania Sophie, married Henry the Mild, Duke of Brunswick-Lüneburg Genealogy

Niebla juncosa

Niebla juncosa is a fruticose lichen that grows on rock, stony soil and sand along the Pacific Coast of Baja California from Punta Banda to Morro Santo Dominogo. The epithet, juncosa is in reference to the thallus divided into rush-like branches, the stems of the flowering plant genus Juncus. Niebla juncosa is distinguished by the thallus divided into sublinear subterete branches with a common attachment base; the species recognized by containing divaricatic acid, with triterpenes. The cortex surrounds a fistulose to subfistulose medulla, varying from 55–175 µm thick thinner on the branches where a change in thickness appears related fragmentation branchlets that break off from primary branches. Two varieties recognized. Variety juncosa has primary branches with entire margins and with secondary branchlets that develop along the upper side of a primary branch. Variety juncosa common on rocks but on sand under Yucca valida in the southern part the Baja peninsula, on stony ground in the terricolous Niebla communities in the transition zone from chaparral to desert scrub on mesas above Punta Baja.

Variety spinulifera differs by primary branches with wavy margins and with spinule-like branchlets that spread in various directions from the primary branch, the spinule branchlets appear to break where attached to the primary branch, appearing to leave a rudimentary lobe. This variety is distributed throughout the range of the species. Niebla juncosa was first recognized from thalli growing on a ridge 300–400 meters in elevation south of Punta Negra, between Punta Rocosa and Punta Prieta. Various outcroppings of rock piles along the ridge were discovered to have a rich assemblage of Niebla species, represented by various morphological and chemotypes; the associated species include large bushy thalli of Niebla homaleoides, Niebla infundibula, Niebla josecuervoi, Niebla sorocarpia, Niebla turgida, other related species with short tufts of branches. Niebla juncosa has been included under a broad species concept of Niebla homalea. Under the broad species concept, the morphological differences are seen as environmentally induced variation, the chemical differences as chemo-syndromes.

Niebla juncosa in Index FungorumWorld Botanical Associates, Niebla juncosa, retrieved 26 Dec 2014, Lichen Flora of the Greater Sonoran Desert: Book Review, Richard Spjut, web page, retrieved 26 Dec 2014,