Naples and Pointless topology: Difference between pages
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'''Pointless topology''' (also called '''point-free''' or '''pointfree''' topology) is an approach to [[topology]] which avoids the mentioning of points. A traditional [[topological space]] consists of a [[set]] of [[point (topology)|points]], together with a system of [[open set]]s. These open sets form a [[lattice (order)|lattice]] with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the [[category theory|category]] of pointless topological spaces, also called ''[[complete Heyting algebra|locales]]'', as an extension of the category of ordinary topological spaces. Some proponents claim that this new category has certain natural properties which make it preferable. Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between [[sober space]]s and spatial locales, are to be found in the article on [[Stone duality]]. |
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:''Alternate uses: See [[Naples (disambiguation)]]'' |
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Formally, we define a ''[[complete Heyting algebra|frame]]'' to be a [[lattice (order)|lattice]] ''L'' in which every (even infinite) subset {''a''<sub>i</sub>} has a supremum V''a''<sub>''i''</sub> such that |
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{{ITdot|Naples}} |
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'''Naples''' ([[Italian language|Italian]] ''Napoli'', [[Neapolitan]] ''Napule'', from [[Greek language|Greek]] Νέα Πόλις - ''Néa Pólis'' - meaning "New City") is the largest city in southern [[Italy]] and capital of [[Campania]] Region. The city has a population of about 1 million, and together with its suburbs, the metropolitan area has 3.7 million inhabitants (''Neapolitans''). It is located just halfway between the [[Vesuvius]] volcano and another unrelated volcanic area, the [[Campi Flegrei]]. |
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:<math>b \wedge \left( \bigvee a_i\right) = \bigvee \left(a_i \wedge b\right)</math> |
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It is rich in [[history|historical]], [[art]]istic and [[culture|cultural]] traditions and [[gastronomy]]. Neapolitan is a [[language]] in its own right, known in Naples as [[Napulitano]]. |
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for all ''b'' and all sets {''a''<sub>''i''</sub>} of ''L''. These frames, together with lattice homomorphisms which respect arbitrary suprema, form a category; the ''opposite'' category of the category of frames is called the category of ''locales'' and generalizes the category of topological spaces. The reason that we take the opposite category is that every continuous map ''f'' : ''X'' → ''Y'' between topological spaces induces a map between the lattices of open sets ''in the opposite direction'': every open set ''O'' in ''Y'' is mapped to the open set ''f''<sup> -1</sup>(''O'') in ''X''. |
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==History== |
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[[image:naples.jpg|right|frame|Gulf of Naples]] |
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It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems. |
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The city was founded by inhabitants of the [[Greek colony]] of [[Cuma]], around the [[8th century BC|eighth century B.C.]]. For this reason it was named '''Neapolis''' (from [[Greek language|Greek]], meaning New City). Its buildings, museums and even the language spoken by natives bear traces of all periods of its history, from its Greek birth, until the present day. Although conquered by the Romans in the 4th century BC, it long retained its Greek culture. |
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While many important theorems in point-set topology require the [[axiom of choice]], this is not true for their analogues in locale theory. |
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This can be useful if one works in a [[topos]] which doesn't have the axiom of choice. |
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The concept of "product of locales" diverges slightly from the concept of "[[Product_topology|product of topological spaces]]", and this divergence has been called a disadvantage of the locale approach. |
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Others claim that the locale product is more natural and point to several of its "desirable" properties which are not shared by products of topological spaces. |
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See also: [[Heyting algebra]]. A locale is a [[complete Heyting algebra]]. |
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It was in Naples, in the 'Castel dell'Ovo' (Castle of the Egg), that [[Romulus Augustus]], the last [[Roman Emperors|Emperor]] of the [[Western Roman Empire]], was imprisoned after being deposed in [[476]]. In the [[6th century|sixth century]], Naples was conquered by the [[Byzantine Empire|Byzantines]] during the attempt of [[Justinian I]] to recreate the Roman Empire, and was one of the last duchies to fall in [[Normans|Norman]] hands in [[1039]], as they founded the Kingdom of [[Sicily]]. |
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==References== |
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[[Frederick II, Holy Roman Emperor|Frederick II Hohenstaufen]] founded its [[university]] in [[1224]]. In [[1266]] Naples and the kingdom of Sicily were assigned by [[Pope Clement IV]] to [[Charles of Anjou]], who moved the capital from [[Palermo]] to Naples. In [[1284]] the kingdom was split in two parts, with an Aragonese king ruling the island of Sicily and the Angevin king ruling the mainland portion; while both kingdoms officially called themselves the Kingdom of Sicily, the mainland portion was commonly referred to as the Kingdom of Naples. This kingdom was much larger than just the city; it covered about the southern third of the boot of the Italian peninsula. |
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* P. T. Johnstone: ''The point of pointless topology''. Bulletin American Mathematical Society, 8(1):41--53, 1983. |
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The two parts would stay separate until [[1816]], when they would form the kingdom of [[Two Sicilies]]. The two kingdoms were united under [[Spain|Spanish]] rule [[1501]], until [[1715]], when Naples became [[Austria]]n until [[1734]]. Under the enlightened [[House of Bourbon|Bourbon]] monarch Charles, king of both Sicilies (''Utriusque Siciliarum'') (later known as [[Charles III of Spain]]), gained independence. |
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In [[1799]], a Jacobin revolution (backed by the French Army) gave birth to a short-lived [[Parthenopaean Republic|republic]] (January - June 1799). |
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[[Category:Category theory]][[Category:General topology]] |
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In [[1861]], the kingdom was conquered by the [[Giuseppe Garibaldi|Garibaldines]] and was handed over to the king of [[Sardinia]]. In October [[1860]] a plebiscite sanctioned the end of the kingdom of Sicily and the birth of the state of [[Italy]]. |
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[[es:Topología sin puntos]] |
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The opening of the funicular railway to [[Mount Vesuvius]] was occasion to the writing of the famous song Funiculì Funiculà, one more song in the centuries long tradition of Neapolitan songs. Many Neapolitan songs are also famous outside of Italy, as for example "[[O Sole Mio]]", "Santa Lucia" and "Torna a Surriento". |
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On [[April 7]], [[1906]] nearby [[Mount Vesuvius]] erupted, devastating [[Boscotrecase]] and seriously damaging Ottaviano. In [[1944]] the activity closed with a spectacular and devastating eruption; images from this eruption were used in the film ''[[The War of the Worlds (1953 movie)|The War of the Worlds]]''. |
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It is still well connected to [[Sicily]] and [[Palermo]]. Naples has an important port that connects it, for example, to [[Cagliari]], [[Genoa]] and [[Palermo]]. Naples has good ferry connections to nearby islands and [[Sorrento]], and fast rail connections to [[Rome]] and the south. It is famous for the light railway [[Circumvesuviana]]. |
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Organised crime is deep-rooted in Naples. The [[Camorra]], the feuding Neapolitan gangs and families, have a long history and are now more of a threat in Italy than the Sicillian-based [[Mafia]]. During 2004 over 120 people died in Naples in Camorra killings, many of the deaths drug-trade related. |
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==Food and drink== |
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Naples is by tradition the home of [[pizza]], specifically it is the birthplace of the Pizza Margherita, which traditionally is made with mozzarella, tomato and basil - representing the red, white, and green of the Italian flag. The pizza was created as homage to Queen Margherita on a vist to the city. La vera pizza (true pizza) should be made in a wood burning oven similar to a [[Tandoori oven]]. There you can find many kinds of Pizza with differents ingredients . |
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Napoli is famous for its excellent pasta dishes, where spaghetti is often served with sugo Di pomodoro, which is an original italian tomato sauce which gets its full flavour from sun-ripe Campanian tomatoes. Another excellent Campanian dish found in Naples is melanzane alla parmigiana , which is fried slices of [[aubergine]] (eggplant) gratined with tomato sauce and [[parmesan]] cheese. Often you can get another version of melenzane alla parmigiana with an addition of [[mozzarella]] cheese. |
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In Naples you can get several kinds of pastry too. The most famous one is, perhaps, the [[babà|baba']], followed by choux and the easter neapolitan cake: the [[pastiera]]. |
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The babà (also known as "savarin") is a mushroom-shaped piece of leavend sweet paste, soaked with an orange flavoured mixture of [[rum|ron|ruhm]] and water. |
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Choux is a small "bubble" of leavend paste stuffed with light cream, usually coffee or chocolate flavored. |
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The pastiera is a cake with plenty of complicated receipts, depending in which county it's been prepared. In it there are usually: annealed, grain, eggs, and, sometimes, cream. But someone adds to it boiled rice and/or spaghetti boiled in milk. |
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Neapolitans also claim that the best espresso coffee in the world is made in their town thanks to special kind of Neapolitan air and water. Naples is also famous for its [[Neapolitan ice cream|gelato]]. Gelato is not ice cream, but rather iced milk. |
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==Tourist attractions== |
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[[Image:CastelDellOvo.jpg|right|thumb|320px|Castel dell'Ovo]] |
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Naples itself is less visited than some of the surrounding attractions. There are, however, many attractions within the city. La Villa Comunale (formerly a royal park) has been refurbished and stretches along the seafront in the smarter western end of the city. It contains an aquarium which is possibly Europe's oldest and is favoured by the locals for family walks on Sunday mornings. The [[Museo Archeologico Nazionale Napoli]] contains a large collection of Roman artifacts from [[Pompeii]] and [[Herculaneum]] as well as the [[Farnese Marbles]], some of the greatest surviving Roman statues; The Museo Nazionale di Capodimonte contains art collections including work by [[Michelangelo]], [[Raffaello Santi|Raphael]], [[Botticelli]] and [[Caravaggio]]. Naples is the home of the [[San Carlo|Teatro di San Carlo]], the oldest active [[opera house]] in [[Europe]], which opened its doors on [[November 4]], [[1737]]. There are also a number of churches, royal palaces and Castles within and around the city. |
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===Under Naples=== |
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Guided tours operate around the Stratification of Naples which shows the city through the layers laid down across history. Subterranean Naples consists of old Greco-Roman reservoirs dug out from the soft [[tuff|tufo stone]] on which, and from which, the city is built. You can visit approximately one kilometer of the many kilometers of tunnels under the city. There are also large catacombs in and around the city. |
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===Also in Naples=== |
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Naples is the site of three major [[military]] bases. [[Naval Support Activity Naples]], located in [[Capodichino]] is a major [[US Navy]] base which is responsible for the support and control of US Naval assets in the [[6th Fleet]] area of responsibility, and [[Bagnoli]], known as [[Joint Force Command South]] (formerly [[AFSOUTH]], many Sailors still call it this) is a major [[NATO]] base, which is responsible for the coordination of NATO forces in the south European Region. There is also the [[Support Site]], which consists mostly of housing and personnel support facilities, located in [[Gricignano di Aversa]]. |
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[[Capodichino]] is the site of the Naples International Airport. |
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===Postcodes of Naples=== |
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80121 [[Chiaia]]<br> |
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80122 [[San Ferdinando]]<br> |
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80123 [[Posillipo]]<br> |
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80124 [[Bagnoli]]<br> |
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80125 [[Bagnoli]] <br> |
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80125 [[Rione Flegreo]]<br> |
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80126 [[Pianura]], [[Soccavo]]<br> |
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80127-80129 [[Vomero]]<br> |
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80128 [[Arenella]] <br> |
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80131 [[Stella (Naples)|Stella]] <br> |
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80131 [[Arenella]]<br> |
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80132 [[Chiaia]] and [[Montecalvario]]<br> |
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80133 [[Porto (Naples)|Porto]]<br> |
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80134 [[Montecalvario]]<br> |
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80135 [[Avvocata]]<br> |
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80136 [[San Lorenzo (Naples)|San Lorenzo]], [[rione Sanità]]<br> |
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80136 [[Stella (Naples)|Stella]], [[località Capodimonte]]<br> |
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80137 [[San Carlo all'Arena]], [[Stella (Naples)|Stella]]<br> |
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80138 [[San Giuseppe]]<br> |
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80139 [[San Lorenzo (Naples)|San Lorenzo]]<br> |
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80141 [[Vicarìa]]<br> |
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80142 [[Mercato]] and [[Pendino]]<br> |
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80143 [[Poggioreale]] and [[Zona industriale]]<br> |
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80144 [[San Pietro a Patierno]], [[Scampìa]], [[Secondigliano]]<br> |
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80145 [[Chiaiano]], [[Miano]], [[Piscìnola]]<br> |
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80146 [[San Giovanni a Teduccio]]<br> |
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80147 [[Barra (Naples)|Barra]], [[Ponticelli]] |
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===Around Naples=== |
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The islands of [[Procida]], famously used as the set for much of [[il Postino]], [[Capri]] and [[Ischia]] can all be reached quickly by Aliscafi (twin-hulled ferries). [[Sorrento]] and the [[Amalfi Coast]] are situated south of Naples. The Roman ruins of [[Pompeii]] and [[Herculaneum]] (destroyed in the A.D. 79 eruption of Vesuvius) are also nearby. |
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===Sporting Naples=== |
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Naples is the home of the underachieving soccer team [[Napoli]]. With the help of [[Diego Maradona]], they achieved rare success in 1987 by winning the [[scudetto]]. In 2004 the football team was declared bankrupt and has been subsequently reborn into the lower division of Serie C1 as 'Napoli Soccer'. |
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===The Neapolitan diaspora=== |
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Naples has seen many of its children spread throughout the world, setting up '[[Little Italy|Little Italies]]' in many countries. The majority of these Neapolitans who left Italy went to the Americas, especially the United States, Canada, Brazil, and Argentina. |
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==Famous Neapolitans== |
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* [[Eduardo Bennato]] also known as Joe Sarnataro |
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* [[Eugenio Bennato]] |
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* [[Gian Lorenzo Bernini]] |
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* [[Giordano Bruno]] |
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* [[Renato Carosone]] |
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* [[Enrico Caruso]] |
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* [[Francesco Cilea]] |
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* [[Domenico Cimarosa]] |
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* [[Benedetto Croce]] |
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* [[Eduardo De Filippo]] |
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* [[Peppino De Filippo]] |
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* [[Enrico De Nicola]] |
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* [[Ruggero Leoncavallo]] |
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* [[Giovanni Leone]] |
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* [[Giovanni Paisiello]] |
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* [[Nicola Salerno]], also known as Nisa |
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* [[Domenico Scarlatti]] |
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* [[Matilde Serao]] |
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* [[Totò]] |
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* [[Massimo Troisi]] |
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* [[Giambattista Vico]] |
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* [[Bud Spencer]] born Carlo Pedersoli |
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* [[Luciano de Crescenzo]] Engineer, Writer, Philosopher |
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* [[Massimo Troisi]] Actor, Director |
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* [[Pino Daniele]] Singer |
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* [[Nino Taranto]] Actor |
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* [[Vincenzo Salemme]] Actor, Director |
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* [[Matilde Serao]] Poet, Journalist (Born in Greek) |
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* [[Salvatore di Giacomo]] Writer, Poet |
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==See also== |
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* [[Monarchs of Naples and Sicily]] |
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== External links== |
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*[http://www.compart-multimedia.com/virtuale/us/napoli/naples.htm Naples, Italy] Virtual reality movies and photo gallery |
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* [http://www.napoli.com/english Around Naples] |
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* [http://www.comune.napoli.it City Council's website] |
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* [http://www.vacation-amalfi.com/excursions/Napoli.html Naples for tourists] |
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* [http://www.inaples.it/eng/ Tourist board website] |
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* [http://earthobservatory.nasa.gov/Newsroom/NewImages/images.php3?img_id=5207 Satellite image of Naples and Vesuvius] at [[NASA]]'s [[Earth Observatory]] |
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* [http://www.verapizzanapoletana.org/vpn/charter.html Association of the Verrace Pizza Napoletana (The True Pizza Society)] |
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* [http://www.marketplace.it/museo.nazionale/ Museo Archeologico Nazionale Napoli (National Archaeological Museum)] |
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* [http://www.channel4.com/sport/football_italia/napoliwatch.html Napoliwatch - reports in English of all Napoli Soccer's matches] |
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[[Category:Coastal cities of Italy]] |
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[[Category:Naples|*]] |
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[[Category:Cities in Campania]] |
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[[Category:Magna Graecia]] |
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[[da:Napoli]] |
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[[de:Neapel]] |
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[[es:Nápoles]] |
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[[fr:Naples]] |
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[[it:Napoli]] |
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[[he:נאפולי]] |
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[[ko:나폴리]] |
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[[nl:Napels]] |
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[[ja:ナポリ]] |
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[[no:Napoli]] |
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[[pl:Neapol]] |
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[[ro:Neapole]] |
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[[simple:Naples]] |
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[[fi:Napoli]] |
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[[sv:Neapel]] |
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Revision as of 20:15, 13 July 2005
Pointless topology (also called point-free or pointfree topology) is an approach to topology which avoids the mentioning of points. A traditional topological space consists of a set of points, together with a system of open sets. These open sets form a lattice with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces. Some proponents claim that this new category has certain natural properties which make it preferable. Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between sober spaces and spatial locales, are to be found in the article on Stone duality.
Formally, we define a frame to be a lattice L in which every (even infinite) subset {ai} has a supremum Vai such that
for all b and all sets {ai} of L. These frames, together with lattice homomorphisms which respect arbitrary suprema, form a category; the opposite category of the category of frames is called the category of locales and generalizes the category of topological spaces. The reason that we take the opposite category is that every continuous map f : X → Y between topological spaces induces a map between the lattices of open sets in the opposite direction: every open set O in Y is mapped to the open set f -1(O) in X.
It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems. While many important theorems in point-set topology require the axiom of choice, this is not true for their analogues in locale theory. This can be useful if one works in a topos which doesn't have the axiom of choice. The concept of "product of locales" diverges slightly from the concept of "product of topological spaces", and this divergence has been called a disadvantage of the locale approach. Others claim that the locale product is more natural and point to several of its "desirable" properties which are not shared by products of topological spaces.
See also: Heyting algebra. A locale is a complete Heyting algebra.
References
- P. T. Johnstone: The point of pointless topology. Bulletin American Mathematical Society, 8(1):41--53, 1983.