History.

The Inter-City Rapid Transit Company was an Ohio interurban company, which began operations in 1930 as it had purchased its route from the Northern Ohio Traction & Light Company. It remained in operation till 1940.  The use of "Inter-City" was reborn in the United Kingdom: A daily train of that name was introduced in 1950, running between the cities of London and Birmingham. This usage can claim to be the origin of all later usages worldwide.  In 1966 British Rail introduced the brand InterCity for all of its express train routes, and in 1986 the term was adopted by the InterCity sector of British Rail. Following the privatisation of the railways in Great Britain, the term is no longer in official use there although many people still refer to fast long-distance services as InterCity trains. The brand still exists though, and belongs to the Department for Transport. In West Germany, the Deutsche Bundesbahn first used the name (then written "Intercity") in 1968, denoting special first-class services on the F-Zug train network. Many of the Class VT 11.5 diesel multiple units formerly used on the TEE network were converted for early Intercity services.  In Switzerland, the InterCity brand replaced SwissExpress in the 1982 schedule.  In Norway, "intercity" (later also written "InterCity") trains were introduced in 1975 on the Vestfold Line (the Oslo-Skien service), later also on the Østfold Line (Oslo-Halden). They were (relatively) fast trains on distances up to 2–3 hours. Today, the name is used not on the trains, but on the main lines from Oslo to Skien, Lillehammer, and Halden – and also on the Ringerike Line, which is under construction from Oslo to Hønefoss.  An international variant of InterCity, EuroCity (EC), was introduced in May 1987. EuroCity trains consist of high-standard, air conditioned coaches, are run by a variety of operators, and are usually subject to on-board border controls. For example, EuroCity trains running in Germany can be made up by rolling stock of either the SBB (Switzerland), ÖBB (Austria), SNCF (France), and less commonly by stock of the Czech ČD and Hungarian MÁV railways.

InterCity by country.

Austria.

The Austrian Federal Railways (ÖBB) have operated IC services since 1991. However, contrary to most other countries, these are often little more than regional rail, as most long-distance, high-standard trains in Austria are likely to be EuroCity (EC) services, even when not leaving the Austrian borders (named "ÖBB-EuroCity" until 2011). Modernised stock of Eurofima coaches is used under the brand name "ÖBB-InterCity" (OIC) mainly on the Austrian Western and Southern Railways from Vienna to Salzburg and Villach.  The ÖBB also deployed electric multiple unit trains, from 2006 also three ICE T (Class 411, named "ÖBB 4011") trainsets in cooperation with the Deutsche Bahn, currently running from Vienna to Frankfurt via Linz and Passau. ÖBB high-speed rail service is provided by Railjet (RJ) trains.  Since 2011, there is an hourly express train service on the Western Railway operated by WESTbahn, the only long-distance competitor of the ÖBB.  IC-Services: The InterCity service from Vienna to Salzburg is going to be expanded for an hourly service to Landeck via Innsbruck by December 2008. Also, the service from Vienna to Graz is going to operate hourly by December 2008.  InterCity stops in Austria:

Belgium.

InterCity trains link all major cities of Belgium. Some of them serve also destinations outside the country. The IC between Liège and Brussels travels at 200 km/h on the HSL 2. However, because of the density of the train network with many connecting lines and the many small- and medium-sized cities in Belgium, most IC services also call at smaller stations or continue as local trains on branch lines.

Croatia.

InterCity trains in Croatia mainly serve domestic routes from Zagreb (capital) to Split and from Zagreb to Osijek except one international train connecting Zagreb and Vinkovci. Speeds are up to 160 km/h between Slavonski Brod and Vinkovci. The coaches used are similar to the ones used in Germany by DB.

Czech Republic.

In the Czech Republic, the IC or Express trains service the following routes:  Most of the IC are served by trains of state-owned operator České dráhy with speeds up to 160 km/h but they have been converted into category Express in 2012 timetable. The other is served by trains of private operator RegioJet with speed up to 160 km/h. No surcharge is applied to IC trains but RegioJet runs reservation compulsory trains. Third private passenger operator LEO Express introduced express InterCity train service between Prague and Ostrava in November 2012, with speeds up to 160 km/h.  The state-owned operator České dráhy also serves line Prague–Ostrava (with some connections extended to Bohumín and Františkovy Lázně) with Pendolino trainsets under designation ""SuperCity"", which conforms to IC standard. A compulsory reservation is applied on these trains.

Denmark.

The Intercity network of the Danish State Railways consists of IC trains and their faster version, "Lyntog" (Lightning Train), which is identical but with fewer stops. Each train type operates hourly between the eastern terminus at Copenhagen and westwards to Odense–Århus–Ålborg, and less frequently to alternative destinations in Jutland. These are run by IC3 diesel materiel since most of the network is not electrified. There are also electrical IC trains run by IR4s in an hourly schedule from Copenhagen westwards to Odense and alternately Esbjerg/Sønderborg. This means during most of the day there are three trains an hour between Copenhagen and Odense. Quite unusual in the world, some trains will consist of both electrically and diesel-powered units coupled together. Being the only option for long-distance and some short-distance travel, there is no surcharge for IC and Lyntog. They have a maximum speed of 180 km/h. Additionally, there are a few IR trains during Friday and Sunday peak hours between Copenhagen and Århus. These are locomotive-run and have bilevel cars. The IC3 trains are planned to be replaced by new IC4 trains, originally in 2001. They first ran with passengers in 2008, but haven't nearly replaced the IC3 yet. It is now planned to give the IC3 stock a 10-year life extension, and to eventually scrap IC4 trains, due to the plague of poor reliability. IR4 stock (Electric version of IC3) will release IC3 stock from the Esbjerg Line, allowing some IC3 trains to replace IC4 trains.

Finland.

In Finland, VR has operated InterCity trains between major Finnish cities since August 1988. The first routes were Helsinki–Vaasa and Helsinki–Imatra, which later expanded to all major cities and include for instance Helsinki–Tampere–Oulu–Rovaniemi, Helsinki–Turku, Helsinki–Iisalmi and Helsinki–Joensuu. InterCity trains have become the standard as the Finnish long-distance rail travel mainstay, and their predecessors, blue-carriaged express trains, are being withdrawn off the schedules. The train tickets include an additional surcharge compared to ordinary express trains: 17 to 27% depending on journey length.  VR operates both ordinary InterCity trains (IC) and entirely double-decker trains (IC2). An ordinary IC train usually consists of 3 to 4 double-decker cars and 3 to 5 ordinary IC cars. In addition the train has a restaurant car. IC2 trains consist only of double-decker cars and have no separate restaurant car with a sales trolley moving about in the train. However, due to high demand, VR ordered double-decker restaurant cars from Transtech in early 2011. The first carriages should be operational in 2013. All IC-trains are smoke-free.  The trains run at a maximum speed of 140–160 km/h for single-decker trains and up to for double-deckers, which was VR's aim in 1988. Only the "Pendolino" and "Allegro" trains are faster than InterCity trains in Finland.

France.

In 2006, the SNCF rebranded Corail, Téoz and Lunéa services as 'Intercités', a brand for all of their 'classic' services day and night.

Germany.

In Germany, the InterCity network was launched in 1971 to accompany and eventually replace the Trans Europ Express trains. At first, IC services were first-class only, often using TEE stock and the then-new Class 103 locomotives. Trains ran bi-hourly. DB paid a royalty fee to BR for many years for the use of the brand name.  In 1978, it was decided to expand the IC network to services with both first and second class, and so the new scheme, called IC <nowiki>'79</nowiki> was launched in 1979 with the motto ""Jede Stunde, jede Klasse"" ("every hour, every class") to emphasize its new structure. Large numbers of air-conditioned open coach cars, the "Bpmz 291", were built for InterCity services, which at first were using the TEE colour scheme. In 1985, with many of the TEE trains gone and the introduction of the InterRegio, the network was expanded again, now covering virtually any major city of then-West Germany. It faced further changes after the German reunification and the introduction of the InterCityExpress in the early 1990s.  Today, after the abolition of the InterRegio in 2002, most long-distance connections in Germany are either IC or ICE trains; they most commonly offer at least bi-hourly service. Maximum speed for an IC is 200 km/h.

Greece.

In Greece, InterCity trains operated by privately owned TrainOSE run in the following routes:

Hungary.

Intercity train services in Hungary:  The Hungarian intercity trains are operated by MAV-START, the Hungarian railway company.

Ireland.

In the Republic of Ireland, Iarnród Éireann introduced the brand name InterCity in 1984, replacing the previous name of "Mainline", which had been introduced in 1976. Initially applied to services operated by British Rail Mark 3 trains, it was later extended to include all services not part of the Dublin Suburban Rail network. Today the brand encompasses services between Dublin and Cork, Galway, Limerick, Tralee, Waterford, Sligo, Westport, Rosslare Europort, Ballina, and Ennis, as well as some regional services. A new InterCity logo was introduced in 2006, though the vast majority of rolling stock bore the original script logo and orange-tan livery until the final Mark 3 set was withdrawn in 2009. Since Ireland completed its replacement of the old Mark 2 and Mark 3 stock in 2012, it has one of the most modern InterCity fleets in the world.  Northern Ireland Railways and Iarnród Éireann both formerly operated trains on the Dublin–Belfast line under the InterCity brand, however this was replaced with the revived Enterprise brand name upon the introduction of the De Dietrich Ferroviaire rolling stock in 1997. This, coupled with the subsequent withdrawal of most coaching stock bearing the logo and the rebrand to the Translink name, means that the InterCity brand has largely disappeared from Northern Ireland Railways.

Italy.

In Italy, InterCity trains constitute a capillary network that links the main cities across the peninsula. There are over 80-day (Intercity) and night (Intercity Notte) services. The major north–south connections are Turin-Lecce (night trains only), Milan-Bari, Bolzano/Bozen-Bari, Venice-Bari, Milan-Rome and Trieste-Naples. In the south, Intercity connections consist of Rome-Palermo, Rome-Bari, Rome-Naples and Rome-Reggio di Calabria. In the north, Intercity connections consist of Turin-Genoa, Milan-Genoa, Milan-Livorno and Venice-Florence.  The fleet consists of first and second class coaches with open-seat or six-seat compartments. Night trains include sleeper coaches of three types: bunk beds for four travellers, two-bed "double" sleeper and one-bed "single" sleeper. There are, however, no showering facilities on any Intercity Notte trains. The newer coaches, called InterCity Plus, have renovated interiors with sockets for PC or mobile phones added to each compartment. However, Excelsior cabins only shown on the Rome To Sicily night train claim to have their own toilet and shower in the cabin, information from Trenitalia web pages:  The Excelsior coaches, available on some night trains offer services comparable to those of a high-level hotel, every vehicle available cabins : • Single: a cabin with a bed, • Double rooms : one twin cabin, • Doubles: a large suite with double bed with all the amenities. Each cabin Excelsior E4 instead offers 4 beds for travel from 1 to 4 persons : Single: a cabin with a bed, Doubles : one twin cabin, Triple : a three-bed cabin Quadruple : a four-bed cabin All cabins are equipped with shower and wash basin with private toilet. High cab there is a reception with bar service. Excelsior single and double cabins are available from all sales channels, while the triple and quadruple only at the ticket and authorized travel agencies. When travelling in these coaches you must have a personal identification document.

Netherlands.

Intercity trains are very common in the Netherlands, but may differ from each other. While some intercity trains only call at larger stations, some other may also call at smaller stations or at all intermediate stations along a short stretch of its route. Hence, some stations may be served by one intercity train while another may pass it. For example, the Intercity trains between Arnhem Central station and Utrecht Central station alternately serve Driebergen-Zeist or Veenendaal-De Klomp (and both when the frequency is lowered from 4tph to 2tph in the evening). Often, intercity trains run as local trains along the very end of its route. For example, the Nijmegen to Den Helder intercity runs as regular intercity until Alkmaar, serving larger stations only, but running as local train between Alkmaar and Den Helder, serving all intermediate stations.  The average distance between station is smaller than most IC trains in most other countries, but frequencies are higher. All lines are served at least with 2tph, but many busy stretches are served have 4tph. From Utrecht Central there are even two different services to Amsterdam, one that serves the central station and two others and another that serves the south station and the airport. From December 2017 the frequency between Amsterdam, Utrecht and Eindhoven will be further increased to 6tph.  Rolling stock used mostly for intercity services in the Netherlands is VIRM. DD-AR trains have been converted to be used as "New Intercity Double deckers" and have been introduced to parts of the intercity network. Also, older ICM (formerly known as Koploper) EMUs are frequently seen on intercity routes. ICRm coaches, pulled by a TRAXX locomotive, can be seen on international stretches and the Intercity Direct.

Poland.

In Poland IC trains operated by PKP Intercity S.A. up to 2009 served the following routes:  Apart from a single railway line (line nr 4, a.k.a. Centralna Magistrala Kolejowa–Central Railway Route), average speeds are much lower.  These trains mostly use the locomotive named Eurosprinter (EU44), EP09, sometimes EP08.  In 2009 IC category was renamed to Express InterCity (EIC for short). Since December 2014 IC trains are regular express trains with common pricing offers as TLK trains, also operated by PKP Intercity, mainly utilizing Pesa Dart (ED161) and Stadler FLIRT (ED160) electric multiple units. In rare cases trains are served with refurbished cars pulled by EU07, EP07 and EP08 locomotives.

Portugal.

In Portugal, InterCidades trains operated by state-owned Comboios de Portugal (CP) run in the following routes:

Romania.

Currently InterCity lines link the capital Bucharest to Brașov, Arad, Oradea, Cluj-Napoca, Târgu Mureș, Galați (all once daily), Suceava, Iași (twice daily), Piatra Neamț (once daily), Bacău (4x daily), Ploiești (6x daily), Craiova (twice daily) and Timișoara (once daily). There are also international trains branded as InterCity between the Hungarian capital, Budapest and some Transylvanian cities such as Cluj-Napoca, Târgu Mureș, Brașov, and others. These trains usually have specific names (Harghita, Ady Endre, etc.).  In the case of IC service running partially or totally on electrified lines as well as international IC-s locomotive-hauled trains are used. For some trains running exclusively on unelectrified lines (mainly in northern Transilvania) Siemens Desiro DMU-s are preferred. These trainsets look similar to standard Desiro DMU-s used for regional trains but have a different interior design with more comfortable seating.  All IC trains use 1st and 2nd class coaches with air conditioning, having automatic doors and a higher level of comfort than InterRegio trains. In order to compete with regional airlines, CFR Călători introduced special Business class coaches on the IC trains running between Oradea and Bucharest and Timișoara and Bucharest. The new class can be divided into two slightly different premium service levels, "Business Exclusiv" (1A) and "Business Standard" (1B). All of these coaches have wireless internet and wall sockets in order to permit the use of laptops, 4 channel audio system, bar, Exclusiv further providing larger space per customer, special leather seating and LCD screens for each seat.  Maximum speed is 140 km/h on Bucharest-Campina, 120 km/h on all other lines.

Serbia.

Currently InterCity lines link the capital Belgrade to Novi Sad, Subotica and Prijepolje.

Slovakia.

In the Slovak Republic, InterCity trains run between the capital Bratislava and Košice (some IC trains continued from Bratislava to Vienna). In the past, the names of some of these trains used to be sold for commercial use (IC Šariš named by beer producer, IC Zelmer named by an electronic devices producer, IC Slovenka named by a magazine, IC Mora, IC Gorenje, both named after kitchen appliances producers). In 2012, no IC train had commercial name. All IC trains are named by natural monuments (IC Tatran, IC Kriváň, IC Gerlach, IC Rysy, IC Chopok, IC Ďumbier). All IC trains are subject to compulsory reservation to make comfort as high as possible. The seat reservation is included in the ticket price. The price varies, depending on the day of travel, and the time of reservation (the sooner you buy a ticket, the lower the price).  In the past, there were also 3 other InterCity services: IC 400/407 Donau (Danube) between Bratislava and Vienna, IC 532/533 Rákoczi from Košice to Budapest and IC 536/537 Hornád from Košice to Pécs. On this train passengers could travel without a reservation or surcharge, since it is impossible to use them for domestic transport in Slovakia.  All InterCity trains consist of new comfortable cars which are fully air conditioned.  First class and a restaurant car are available on all IC trains.  Maximum speed between Bratislava and Nové Mesto nad Váhom has been increased in the recent years and is now 160 km/h. On the rest of the route, maximum speed varies from 80 to 140 km/h. There is a large reconstruction under way between Nové Mesto nad Váhom and Žilina. After the reconstruction is finished, it will be possible to increase maximum speed of this part of the route to 160 km/h.

Slovenia.

Intercity trains in Slovenia mainly serve domestic routes, like running from Ljubljana (capital) to Celje, Ptuj, and Maribor. Comfort in Slovenian Intercity trains is not as good as in other countries. Coaches serving main routes are usually old. Some IC trains do not offer first class and restaurant carriage. The type of cars is basically the same as the cars of ÖBB, however, the cars are slightly older with no sign of renovation. Additionally, there are also available older Pendolino electric multiple tilting units ETR 310 (SŽ series 310) labeled as ICS lines connecting Slovenian largest cities, Ljubljana and Maribor and the Mediterranean region of Slovenia, Koper.

Spain.

Renfe operates intercity trains in Spain. It was a long-distance train with high frequency. This denomination was created firstly in 1980. It started operating the electric train (EMU) Renfe 444 series, making the Madrid -Valencia vía Albacete line. It was capable of doing 140 km/h (87 mph).  In 1986 it expanded to Barcelona and Zaragoza, it was capable of doing 160 km/h (99 mph). In 1987 Renfe received new 448 series electric trains, those also being capable of doing 160 km/h. In the main lines this service was combined with fast trains or Talgos. In the less used lines it was a once a day train, as daily fast train. The last line was Irún -Madrid, operated by a 448 series. This service closed in 2008.  In 2012 they opened new lines with the same denomination, making medium and long-distance routes.  The main differences between this line and the other ones is that it has more stops, trains stop in some lines every 15 minutes or less. This service can be divided in 3 categories: -New lines using old Medium distance tracks, but with more stops. -Long-distance routes with intercity fares. -Old MD (Medium distance) lines, some of them are two older MD lines together.

Sweden.

During the 1980s, InterCity denoted trains of the highest standard in Sweden, serving as a fast and comfortable connection between major Swedish cities. Trains used to be set up of the most modern cars, always including a restaurant car. During the 1990s the Swedish State Railways introduced the new X 2000 units which replaced the InterCity trains in this role. Since then, the InterCity trains serve an auxiliary role, calling at smaller stations and providing a cheaper alternative for the costly X 2000. They are set up of modernised cars pulled by Rc6 locomotives with a maximum speed of 160 km/h. They serve the following lines: Apart from the SJ, there are other train operators in Sweden who also run trains of a similar type. Only SJ, however, uses the name "InterCity".

Switzerland.

Swiss InterCity services started in 1982, replacing the Swiss Express on the line Geneva-St. Gallen. There is no surcharge for InterCity services in Switzerland and the rolling stock consists of three types of formations: If needed, two ICN trains can be combined to double the passenger capacity. In addition, several type IV carriages can be attached to an IC 2000 train set on the other side of the locomotive, especially if a group reservation requires extra carriages.  All carriages are air conditioned. Each train formation has first and second class carriages and, in most cases, a restaurant/bistro carriage. The availability of power sockets is constantly improved, sometimes even in second class. The maximal speed of all train types is 200 km/h, except for about half of type IV carriages that have not yet been upgraded and have a maximum speed of 160 km/h.  InterCity trains serve most of Swiss cities and provide direct connections from Zurich and Geneva airports. The InterCity routes within Switzerland according to 2009–2010 schedule include:  as well as the following routes served by tilting ICNs:

Ukraine.

Intercity+ (max. 160 km/h ) train services in Ukraine : There were trains from Kharkiv and Dnipro to Crimea and from Kyiv to Donetsk. They were suspended in 2014, shortly after Ukraine have lost control of some of its territories. Currently, a shortened connection to Donbas is in service: These services are operated by Hyundai Rotem HRCS2, Škoda UZ class 675, KVBZ Ekr1 "Tarpan" trains, and KVBZ "Ukrajina" locomotive-hauled train.  Intercity (70 km/h – 90 km/h) train services in Ukraine :  Intercity and Intercity+ trains are operated by Ukrainian Rapid Railway Company (a branch of Ukrzaliznytsia).

United Kingdom.

The term "Inter-City" was first used by state railway company British Rail in 1966, to brand all its longer-distance, higher-speed services (the hyphen was dropped shortly afterwards, changing the name to "InterCity"). The brand was closely associated with a new design of carriage, the Mark 2 which revolutionised levels of comfort on the system. The system was hugely successful and became one of the world's few profitable public railway services.  Today, Britain's railways having been privatised, InterCity trains in the UK are operated by many different train companies including Abellio Greater Anglia, Avanti West Coast, CrossCountry, Great Western Railway, London North Eastern Railway, East Midlands Railway, Hull Trains and Grand Central. Most of these companies operate their services from one of the many London termini, with CrossCountry being the exception, running from Cornwall to Scotland. If travelling via London it is often necessary to change stations: a journey from Norwich to Cardiff would require a transfer from London Liverpool Street to London Paddington station via the London Underground.  Currently, InterCity trains in United Kingdom use InterCity 125 (Class 43 power car + 8/9 Mark 3 coaches, also named as HST), InterCity 225 (Class 91 + 9 Mark 4 coaches), Class 90 + 8/9 Mark 3 coaches + Driving Van Trailer, Voyager family classes 220, 221 & 222 (4–7 coach DEMUs), class 180 (5 coach DMUs) and class 390 (9 or 11 coach EMUs) on the Intercity lines.  The British Government have named their project to replace the InterCity 125 as the Intercity Express Programme.

InterCity (commonly abbreviated IC on timetables and tickets) is the classification applied to certain long-distance passenger train services in Europe. Such trains (in contrast to regional, local, or commuter trains) generally call at major stations only.

InterCity (zkratka IC) je v České republice a některých dalších zemích druh expresního vlaku pro cestující, který jezdí na vnitrostátních trasách a jsou obdobou mezinárodních vlaků EuroCity. Z hlediska dopravních předpisů Českých drah je vlakem kategorie InterCity i vlak s obchodním označením SuperCity.


Cover in topology.

Covers are commonly used in the context of topology. If the set "X" is a topological space, then a "cover" "C" of "X" is a collection of subsets "U" of "X" whose union is the whole space "X". In this case we say that "C" "covers" "X", or that the sets "U" "cover" "X". Also, if "Y" is a subset of "X", then a "cover" of "Y" is a collection of subsets of "X" whose union contains "Y", i.e., "C" is a cover of "Y" if  Let "C" be a cover of a topological space "X". A subcover of "C" is a subset of "C" that still covers "X".  We say that "C" is an if each of its members is an open set (i.e. each "U" is contained in "T", where "T" is the topology on "X").  A cover of "X" is said to be locally finite if every point of "X" has a neighborhood that intersects only finitely many sets in the cover. Formally, "C" = {"U"} is locally finite if for any "x" ∈ "X", there exists some neighborhood "N"("x") of "x" such that the set is finite. A cover of "X" is said to be point finite if every point of "X" is contained in only finitely many sets in the cover. A cover is point finite if it is locally finite, though the converse is not necessarily true.

Refinement.

A refinement of a cover "C" of a topological space "X" is a new cover "D" of "X" such that every set in "D" is contained in some set in "C". Formally,  In other words, there is a refinement map formula_11 satisfying formula_12 for every formula_13. This map is used, for instance, in the Čech cohomology of "X".  Every subcover is also a refinement, but the opposite is not always true. A subcover is made from the sets that are in the cover, but omitting some of them; whereas a refinement is made from any sets that are subsets of the sets in the cover.  The refinement relation is a preorder on the set of covers of "X".  Generally speaking, a refinement of a given structure is another that in some sense contains it. Examples are to be found when partitioning an interval (one refinement of formula_14 being formula_15), considering topologies (the standard topology in euclidean space being a refinement of the trivial topology). When subdividing simplicial complexes (the first barycentric subdivision of a simplicial complex is a refinement), the situation is slightly different: every simplex in the finer complex is a face of some simplex in the coarser one, and both have equal underlying polyhedra.  Yet another notion of refinement is that of star refinement.

Subcover.

A simple way to get a subcover is to omit the sets contained in another set in the cover. Consider specifically open covers. Let formula_16 be a topological basis of formula_1 and formula_18 be an open cover of formula_1. First take formula_20. Then formula_21 is a refinement of formula_18. Next, for each formula_23, we select a formula_24 containing formula_25 (requiring the axiom of choice). Then formula_26 is a subcover of formula_18. Hence the cardinality of a subcover of an open cover can be as small as that of any topological basis. Hence in particular second countability implies a space is Lindelöf.

Compactness.

The language of covers is often used to define several topological properties related to "compactness". A topological space "X" is said to be  For some more variations see the above articles.

Covering dimension.

A topological space "X" is said to be of covering dimension "n" if every open cover of "X" has a point-finite open refinement such that no point of "X" is included in more than "n+1" sets in the refinement and if "n" is the minimum value for which this is true. If no such minimal "n" exists, the space is said to be of infinite covering dimension.

In mathematics, particularly topology, a cover of a set formula_1 is a collection of sets whose union includes formula_1 as a subset. Formally, if formula_3 is an indexed family of sets formula_4, then formula_5 is a cover of formula_1 if

Pokrytím množiny formula_1 nazýváme takový systém množin formula_2, že formula_3. Jsou-li všechny množiny ze systému formula_2 otevřené, pak hovoříme o "otevřeném pokrytí". Jestliže je část formula_5 systému formula_2 pokrytím množiny formula_1, pak ji nazýváme "podpokrytím" formula_1.


Examples.

The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. Consider for instance the sequence defined by formula_2 and formula_3 This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit "x", then by solving formula_4 necessarily "x" = 2, yet no rational number has this property. However, considered as a sequence of real numbers, it does converge to the irrational number formula_1.  The open interval, again with the absolute value metric, is not complete either. The sequence defined by "x" = is Cauchy, but does not have a limit in the given space. However the closed interval is complete; for example the given sequence does have a limit in this interval and the limit is zero.  The space R of real numbers and the space C of complex numbers (with the metric given by the absolute value) are complete, and so is Euclidean space R, with the usual distance metric. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces. The space C of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C of continuous functions on, for it may contain unbounded functions. Instead, with the topology of compact convergence, C can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.  The space Q of "p"-adic numbers is complete for any prime number "p". This space completes Q with the "p"-adic metric in the same way that R completes Q with the usual metric.  If "S" is an arbitrary set, then the set "S" of all sequences in "S" becomes a complete metric space if we define the distance between the sequences ("x") and ("y") to be, where "N" is the smallest index for which "x" is distinct from "y", or 0 if there is no such index. This space is homeomorphic to the product of a countable number of copies of the discrete space "S".

Some theorems.

A metric space "X" is complete if and only if every decreasing sequence of non-empty closed subsets of "X", with diameters tending to 0, has a non-empty intersection: if "F" is closed and non-empty, for every "n", and, then there is a point common to all sets "F".  Every compact metric space is complete, though complete spaces need not be compact. In fact, a metric space is compact if and only if it is complete and totally bounded. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace "S" of R is compact and therefore complete.  Let be a complete metric space. If is a closed set, then "A" is also complete. Let be a metric space. If is a complete subspace, then "A" is also closed.  If "X" is a set and "M" is a complete metric space, then the set of all bounded functions "f" from "X" to "M" is a complete metric space. Here we define the distance in in terms of the distance in "M" with the supremum norm  If "X" is a topological space and "M" is a complete metric space, then the set "C"("X", "M") consisting of all continuous bounded functions "f" from "X" to "M" is a closed subspace of and hence also complete.  The Baire category theorem says that every complete metric space is a Baire space. That is, the union of countably many nowhere dense subsets of the space has empty interior.  The Banach fixed point theorem states that a contraction mapping on a complete metric space admits a fixed point. The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.  The expansion constant of a metric space is the infimum of all constants formula_7 such that whenever the family formula_8 intersects pairwise, the intersection  is nonempty. A metric space is complete if and only if its expansion constant is ≤ 2.

Completion.

For any metric space "M", one can construct a complete metric space "M′" (which is also denoted as ), which contains "M" as a dense subspace. It has the following universal property: if "N" is any complete metric space and "f" is any uniformly continuous function from "M" to "N", then there exists a unique uniformly continuous function "f′" from "M′" to "N" that extends "f". The space "M"' is determined up to isometry by this property (among all complete metric spaces isometrically containing "M"), and is called the "completion" of "M".  The completion of "M" can be constructed as a set of equivalence classes of Cauchy sequences in "M". For any two Cauchy sequences "x" = ("x") and "y" = ("y") in "M", we may define their distance as  (This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of "M". The original space is embedded in this space via the identification of an element "x" of "M"' with the equivalence class of sequences in "M" converging to "x" (i.e., the equivalence class containing the sequence with constant value "x"). This defines an isometry onto a dense subspace, as required. Notice, however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment.  Cantor's construction of the real numbers is similar to the above construction; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances. The additional subtlety to contend with is that it is not logically permissible to use the completeness of the real numbers in their own construction. Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a field that has the rational numbers as a subfield. This field is complete, admits a natural total ordering, and is the unique totally ordered complete field (up to isomorphism). It is "defined" as the field of real numbers (see also Construction of the real numbers for more details). One way to visualize this identification with the real numbers as usually viewed is that the equivalence class consisting of those Cauchy sequences of rational numbers that "ought" to have a given real limit is identified with that real number. The truncations of the decimal expansion give just one choice of Cauchy sequence in the relevant equivalence class.  For a prime "p", the "p"-adic numbers arise by completing the rational numbers with respect to a different metric.  If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.

Topologically complete spaces.

Note that completeness is a property of the "metric" and not of the "topology", meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval, which is not complete.  In topology one considers "completely metrizable spaces", spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.  Completely metrizable spaces are often called "topologically complete". However, the latter term is somewhat arbitrary since metric is not the most general structure on a topological space for which one can talk about completeness (see the section Alternatives and generalizations). Indeed, some authors use the term "topologically complete" for a wider class of topological spaces, the completely uniformizable spaces.  A topological space homeomorphic to a separable complete metric space is called a Polish space.

Alternatives and generalizations.

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure. This is most often seen in the context of topological vector spaces, but requires only the existence of a continuous "subtraction" operation. In this setting, the distance between two points "x" and "y" is gauged not by a real number "ε" via the metric "d" in the comparison "d"("x", "y") < "ε", but by an open neighbourhood "N" of 0 via subtraction in the comparison "x" − "y" ∈ "N".  A common generalisation of these definitions can be found in the context of a uniform space, where an entourage is a set of all pairs of points that are at no more than a particular "distance" from each other.  It is also possible to replace Cauchy "sequences" in the definition of completeness by Cauchy "nets" or Cauchy filters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in "X", then "X" is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most general situation in which Cauchy nets apply is Cauchy spaces; these too have a notion of completeness and completion just like uniform spaces.

In mathematical analysis, a metric space "M" is called complete (or a Cauchy space) if every Cauchy sequence of points in "M" has a limit that is also in "M" or, alternatively, if every Cauchy sequence in "M" converges in "M".

Metrický prostor je označován jako úplný, pokud v něm každá posloupnost, která je cauchyovská v příslušné metrice, konverguje (v příslušné metrice).


History.

Origin of the term.

The term "speciesism", and the argument that it is simply a prejudice, first appeared in 1970 in a privately printed pamphlet written by British psychologist Richard D. Ryder. Ryder was a member of a group of academics in Oxford, England, the nascent animal rights community, now known as the Oxford Group. One of the group's activities was distributing pamphlets about areas of concern; the pamphlet titled "Speciesism" was written to protest against animal experimentation.  Ryder stated in the pamphlet that "[s]ince Darwin, scientists have agreed that there is no'magical' essential difference between humans and other animals, biologically-speaking. Why then do we make an almost total distinction morally? If all organisms are on one physical continuum, then we should also be on the same moral continuum." He wrote that, at that time in the UK, 5,000,000 animals were being used each year in experiments, and that attempting to gain benefits for our own species through the mistreatment of others was "just'speciesism' and as such it is a selfish emotional argument rather than a reasoned one". Ryder used the term again in an essay, "Experiments on Animals", in "Animals, Men and Morals" (1971), a collection of essays on animal rights edited by philosophy graduate students Stanley and Roslind Godlovitch and John Harris, who were also members of the Oxford Group. Ryder wrote:  In as much as both "race" and "species" are vague terms used in the classification of living creatures according, largely, to physical appearance, an analogy can be made between them. Discrimination on grounds of race, although most universally condoned two centuries ago, is now widely condemned. Similarly, it may come to pass that enlightened minds may one day abhor "speciesism" as much as they now detest "racism." The illogicality in both forms of prejudice is of an identical sort. If it is accepted as morally wrong to deliberately inflict suffering upon innocent human creatures, then it is only logical to also regard it as wrong to inflict suffering on innocent individuals of other species.... The time has come to act upon this logic.  Those who claim that speciesism is unfair to individuals of nonhuman species have often invoked mammals and chickens in the context of research or farming. There is not yet a clear definition or line agreed upon by a significant segment of the movement as to which species are to be treated equally with humans or in some ways additionally protected: mammals, birds, reptiles, arthropods, insects, bacteria, etc. This question is all the more complex since a study by Miralles et al. (2019) has brought to light the evolutionary component of human empathic and compassionate reactions and the influence of anthropomorphic mechanisms in our affective relationship with the living world as a whole: the more an organism is evolutionarily distant from us, the less we recognize ourselves in it and the less we are moved by its fate.

Spread of the idea.

The term was popularized by the Australian philosopher Peter Singer in his book "Animal Liberation" (1975). Singer had known Ryder from his own time as a graduate philosophy student at Oxford. He credited Ryder with having coined the term and used it in the title of his book's fifth chapter: "Man's Dominion... "a short history of speciesism"", defining it as "a prejudice or attitude of bias in favour of the interests of members of one's own species and against those of members of other species":  Racists violate the principle of equality by giving greater weight to the interests of members of their own race when there is a clash between their interests and the interests of those of another race. Sexists violate the principle of equality by favouring the interests of their own sex. Similarly, speciesists allow the interests of their own species to override the greater interests of members of other species. The pattern is identical in each case.  Singer stated from a preference-utilitarian perspective, writing that speciesism violates the principle of equal consideration of interests, the idea based on Jeremy Bentham's principle: "each to count for one, and none for more than one." Singer stated that, although there may be differences between humans and nonhumans, they share the capacity to suffer, and we must give equal consideration to that suffering. Any position that allows similar cases to be treated in a dissimilar fashion fails to qualify as an acceptable moral theory. The term caught on; Singer wrote that it was an awkward word but that he could not think of a better one. It became an entry in the "Oxford English Dictionary" in 1985, defined as "discrimination against or exploitation of animal species by human beings, based on an assumption of mankind's superiority." In 1994 the "Oxford Dictionary of Philosophy" offered a wider definition: "By analogy with racism and sexism, the improper stance of refusing respect to the lives, dignity, or needs of animals of other than the human species."  More recently, animal rights groups such as Farm Animal Rights Movement and People for the Ethical Treatment of Animals have attempted to popularize the concept by promoting a World Day Against Speciesism on June 5.

Arguments against.

Moral community, argument from marginal cases.

Paola Cavalieri writes that the current humanist paradigm is that only human beings are members of the moral community, and that all are worthy of equal protection. Species membership, she writes, is "ipso facto" moral membership. The paradigm has an inclusive side (all human beings deserve equal protection) and an exclusive one (only human beings have that status).  She writes that it is not only philosophers who have difficulty with this concept. Richard Rorty (1931–2007) stated that most human beings – those outside what he called our "Eurocentric human rights culture" – are unable to understand why membership of a species would in itself be sufficient for inclusion in the moral community: "Most people live in a world in which it would be just too risky – indeed, it would often be insanely dangerous – to let one's sense of moral community stretch beyond one's family, clan or tribe." Rorty wrote:  Such people are "morally" offended by the suggestion that they should treat someone who is not kin as if he were a brother, or a nigger as if he were white, or a queer as if he were normal, or an infidel as if she were a believer. They are offended by the suggestion that they treat people whom they do not think of as human as if they were human. When utilitarians tell them that all pleasures and pains felt by members of our biological species are equally relevant to moral deliberation, or when Kantians tell them that the ability to engage in such deliberation is sufficient for membership in the moral community, they are incredulous. They rejoin that these philosophers seem oblivious to blatantly obvious moral distinctions, distinctions that any decent person will draw.  Much of humanity is similarly offended by the suggestion that the moral community be extended to nonhumans. Nonhumans do possess some moral status in many societies, but it generally extends only to protection against what Cavalieri calls "wanton cruelty". Anti-speciesists state that the extension of moral membership to all humanity, regardless of individual properties such as intelligence, while denying it to nonhumans, also regardless of individual properties, is internally inconsistent. According to the argument from marginal cases, if infants, the senile, the comatose, and the cognitively disabled (marginal-case human beings) have a certain moral status, then nonhuman animals must be awarded that status too, since there is no morally relevant ability that the marginal-case humans have that nonhumans lack.  American legal scholar Steven M. Wise states that speciesism is a bias as arbitrary as any other. He cites the philosopher R.G. Frey (1941–2012), a leading animal rights critic, who wrote in 1983 that, if forced to choose between abandoning experiments on animals and allowing experiments on "marginal-case" humans, he would choose the latter, "not because I begin a monster and end up choosing the monstrous, but because I cannot think of anything at all compelling that cedes all human life of any quality greater value than animal life of any quality".

"Discontinuous mind".

Richard Dawkins, the evolutionary biologist, stated against speciesism in "The Blind Watchmaker" (1986), "The Great Ape Project" (1993), and "The God Delusion" (2006), elucidating the connection with evolutionary theory. He compares former racist attitudes and assumptions to their present-day speciesist counterparts. In the chapter "The one true tree of life" in "The Blind Watchmaker", he states that it is not only zoological taxonomy that is saved from awkward ambiguity by the extinction of intermediate forms, but also human ethics and law. Dawkins states that what he calls the "discontinuous mind" is ubiquitous, dividing the world into units that reflect nothing but our use of language, and animals into discontinuous species:  The director of a zoo is entitled to "put down" a chimpanzee that is surplus to requirements, while any suggestion that he might "put down" a redundant keeper or ticket-seller would be greeted with howls of incredulous outrage. The chimpanzee is the property of the zoo. Humans are nowadays not supposed to be anybody's property, yet the rationale for discriminating against chimpanzees is seldom spelled out, and I doubt if there is a defensible rationale at all. Such is the breathtaking speciesism of our Christian-inspired attitudes, the abortion of a single human zygote (most of them are destined to be spontaneously aborted anyway) can arouse more moral solicitude and righteous indignation than the vivisection of any number of intelligent adult chimpanzees!... The only reason we can be comfortable with such a double standard is that the intermediates between humans and chimps are all dead.  Dawkins elaborated in a discussion with Singer at The Center for Inquiry in 2007, when asked whether he continues to eat meat: "It's a little bit like the position which many people would have held a couple of hundred years ago over slavery. Where lots of people felt morally uneasy about slavery but went along with it because the whole economy of the South depended upon slavery."

Animal holocaust.

David Sztybel states in his paper, "Can the Treatment of Animals Be Compared to the Holocaust?" (2006), that the racism of the Nazis is comparable to the speciesism inherent in eating meat or using animal by-products, particularly those produced on factory farms. Y. Michael Barilan, an Israeli physician, states that speciesism is not the same thing as Nazi racism, because the latter extolled the abuser and condemned the weaker and the abused. He describes speciesism as the recognition of rights on the basis of group membership, rather than solely on the basis of moral considerations.

Centrality of consciousness.

"Libertarian extension" is the idea that the intrinsic value of nature can be extended beyond sentient beings. This seeks to apply the principle of individual rights not only to all animals but also to objects without a nervous system such as trees, plants, and rocks. Ryder rejects this argument, writing that "value cannot exist in the absence of consciousness or potential consciousness. Thus, rocks and rivers and houses have no interests and no rights of their own. This does not mean, of course, that they are not of value to us, and to many other painients, including those who need them as habitats and who would suffer without them."

Arguments in favor.

Philosophical.

A common theme in defending speciesism is the argument that humans have the right to exploit other species to defend their own. Philosopher Carl Cohen stated in 1986: "Speciesism is not merely plausible; it is essential for right conduct, because those who will not make the morally relevant distinctions among species are almost certain, in consequence, to misapprehend their true obligations." Cohen writes that racism and sexism are wrong because there are no relevant differences between the sexes or races. Between people and animals, he states, there are significant differences; his view is that animals do not qualify for Kantian personhood, and as such have no rights.  Nel Noddings, the American feminist, has criticized Singer's concept of speciesism for being simplistic, and for failing to take into account the context of species preference, as concepts of racism and sexism have taken into account the context of discrimination against humans. Peter Staudenmaier has stated that comparisons between speciesism and racism or sexism are trivializing:  The central analogy to the civil rights movement and the women's movement is trivializing and ahistorical. Both of those social movements were initiated and driven by members of the dispossessed and excluded groups themselves, not by benevolent men or white people acting on their behalf. Both movements were built precisely around the idea of reclaiming and reasserting a shared humanity in the face of a society that had deprived it and denied it. No civil rights activist or feminist ever argued, "We're sentient beings too!" They argued, "We're fully human too!" Animal liberation doctrine, far from extending this humanist impulse, directly undermines it.  A similar argument was made by Bernard Williams, who observed that a difference between speciesism versus racism and sexism is that racists and sexists deny any input from those of a different race or sex when it comes to questioning how they should be treated. Conversely, when it comes to how animals should be treated by humans, Williams observed that it is only possible for humans to discuss that question. Williams observed that being a human being is often used as an argument against discrimination on the grounds of race or sex, whereas racism and sexism are seldom deployed to counter discrimination.  Williams also stated in favour of speciesism (which he termed 'humanism'), arguing that "Why are fancy properties which are grouped under the label of personhood "morally relevant" to issues of destroying a certain kind of animal, while the property of being a human being is not?" Williams states that to respond by arguing that it is because these are properties considered valuable by human beings does not undermine speciesism as humans also consider human beings to be valuable, thus justifying speciesism. Williams then states that the only way to resolve this would be by arguing that these properties are "simply better" but in that case one would need to justify why these properties are better if not because of human attachment to them. Christopher Grau supported Williams, arguing that if one used properties like rationality, sentience and moral agency as criteria for moral status as an alternative to species-based moral status, then it would need to be shown why these particular properties are to be used instead of others; there must be something that gives them special status. Grau states that to claim these are simply better properties would require the existence of an impartial observer, an "enchanted picture of the universe", to state them to be so. Thus Grau states that such properties have no greater justification as criteria for moral status than being a member of a species does. Grau also states that even if such an impartial perspective existed, it still wouldn't necessarily be against speciesism, since it is entirely possible that there could be reasons given by an impartial observer for humans to care about humanity. Grau then further observes that if an impartial observer existed and valued only minimalizing suffering, it would likely be overcome with horror at the suffering of all individuals and would rather have humanity annihilate the planet than allow it to continue. Grau thus concludes that those endorsing the idea of deriving values from an impartial observer do not seem to have seriously considered the conclusions of such an idea.  Another criticism of animal-type anti-speciesism is based on the distinction between demanding rights one wants and being put into those one may not want. Many people, who are now over 18 but remember their time as minors as a time when their alleged children's rights were legalized torture, doubt if animal rights do animals any good, especially since animals cannot even say what they consider to be horrible. A distinction is made between people who are extrinsically denied their possibility to say what they think by 18 year limits, psychiatric diagnoses based on domain-specific hypotheses, or other constructed laws on one hand, and marginal case humans intrinsically incapable of opining about their situation on the other. The former is considered comparable to racism and sexism, the latter is considered comparable to animals. This extends to questioning and rejecting the very definition of "wanton cruelty". One example that has been pointed out is that since we do not know whether or not animals are aware of death, all ethical considerations on putting animals down are benighted. Advocates of this way of partly accepting speciesism generally do not subscribe to arguments about alleged dehumanization or other legalistic type arguments, and have no problem with accepting possible future encounters with extraterrestrial intelligence or artificial intelligence as equals.  Ayn Rand's Objectivism holds that humans are the only beings who have what Rand called a conceptual consciousness, and the ability to reason and develop a moral system. She stated that humans are therefore the only species entitled to rights. Objectivist philosopher Leonard Peikoff stated: "By its nature and throughout the animal kingdom, life survives by feeding on life. To demand that man defer to the 'rights' of "other" species is to deprive man himself of the right to life. This is 'other-ism,' i.e. altruism, gone mad."  Douglas Maclean agreed that Singer raised important questions and challenges, particularly with his argument from marginal cases. However, Maclean questioned if different species can be fitted with human morality, observing that animals were generally held exempt from morality; if a man were to kidnap and try to kill a woman, most people would be outraged and anyone who intervened would be lauded as a hero, yet if a hawk captured and killed a marmot, most people would react in awe of nature and criticize anyone who tried to intervene. Maclean thus suggests that morality only makes sense under human relations, with the further one gets from it the less it can be applied. Maclean further stated that species membership is used to humanize other people and create concepts such as dignity, respect and the capacity to be treated as something more than creatures driven by survival and reproduction.  The British philosopher, Roger Scruton, regards the emergence of the animal rights and anti-speciesism movement as "the strangest cultural shift within the liberal worldview", because the idea of rights and responsibilities is, he states, distinctive to the human condition, and it makes no sense to spread them beyond our own species. Scruton states that if animals have rights, then they also have duties, which animals would routinely violate, with almost all of them being "habitual law-breakers" and predatory animals such as foxes, wolves and killer whales being "inveterate murderers" who "should be permanently locked up". He accuses anti-speciesism advocates of "pre-scientific" anthropomorphism, attributing traits to animals that are, he says, Beatrix Potter-like, where "only man is vile." It is, he states, a fantasy, a world of escape.  Thomas Wells, while agreeing that humans should have duties towards the natural world, stated that Peter Singer's arguments were incoherent. Wells states that Singer's call for ending animal suffering would justify simply exterminating every animal on the planet in order to prevent the numerous ways in which they suffer, as they could no longer feel any pain. Wells also stated that by focusing on the suffering humans inflict on animals and ignoring suffering animals inflict upon themselves or that inflicted by nature, Singer is creating a hierarchy where some suffering is more important than others, despite claiming to be committed to equality of suffering. Wells also states that the capacity to suffer, Singer's criteria for moral status, is one of degree rather than absolute categories; Wells observes that Singer denies moral status to plants on the grounds they cannot subjectively feel anything (even though they react to stimuli), yet Wells states there is no indication that animals feel pain and suffering the way humans do. Wells thus concludes "The inconvenient topography of sentience, and the hierarchy of interests it implies has to be flattened out, lest the reader conclude that something more sophisticated than hedonic utilitarianism is required."  Robert Nozick notes that if species membership is irrelevant, then this would mean that endangered animals have no special claim.

Religious.

The Rev. John Tuohey, founder of the Providence Center for Health Care Ethics, writes that the logic behind the anti-speciesism critique is flawed, and that, although the animal rights movement in the United States has been influential in slowing animal experimentation, and in some cases halting particular studies, no one has offered a compelling argument for species equality.  Some proponents of speciesism believe that animals exist so that humans may make use of them. They state that this special status conveys special rights, such as the right to life, and also unique responsibilities, such as stewardship of the environment. This belief in human exceptionalism is often rooted in the Abrahamic religions, such as the Book of Genesis 1:26: "Then God said, "Let Us make man in Our image, according to Our likeness; and let them rule over the fish of the sea and over the birds of the sky and over the cattle and over all the earth, and over every creeping thing that creeps on the earth." Animal rights advocates state that dominion refers to stewardship, not ownership. Jesus Christ taught that a person is worth more than many sparrows. But the Imago Dei may be personhood itself, although we humans have only achieved efficiencies in educating and otherwise acculturating humans. Proverbs 12:10 says that "Whoever is righteous has regard for the life of his beast, but the mercy of the wicked is cruel."

Social psychology.

Psychologists have also considered examining speciesism as a specific psychological construct or attitude (as opposed to speciesism as a philosophy). Studies have found that speciesism is a stable construct that differs amongst personalities and correlates with other variables. For example, speciesism has been found to have a weak positive correlation with homophobia and right-wing authoritarianism, as well as slightly stronger correlations with political conservatism, racism and system justification. Moderate positive correlations were found with social dominance orientation and sexism. Social dominance orientation was theorised to be underpinning most of the correlations; controlling for social dominance orientation reduces all correlations substantially and renders many statistically insignificant. Speciesism likewise predicts levels of prosociality toward animals and behavioural food choices.  Some research has found that laypeople are aware of a connection between speciesism and other forms of prejudice; laypeople tend to infer similar personality traits and prejudices from a speciesist as they do from a racist, sexist, or homophobe. However, it is not clear if there is a link between speciesism and non-traditional forms of prejudice such as negative attitudes towards the overweight or towards Christians.  Psychological studies have furthermore argued that people tend to "morally value individuals of certain species less than others even when beliefs about intelligence and sentience are accounted for."

Law and policy.

Law.

The first major statute addressing animal protection in the United States, titled "An Act for the More Effectual Prevention of Cruelty to Animals", was enacted in 1867. It provided the right to incriminate and enforce protection with regards to animal cruelty. The act, which has since been revised to suit modern cases state by state, originally addressed such things as animal neglect, abandonment, torture, fighting, transport, impound standards, and licensing standards. Although an animal rights movement had already started as early as the late 1800s, some of the laws that would shape the way animals would be treated as industry grew, were enacted around the same time that Richard Ryder was bringing the notion of Speciesism to the conversation. Legislation was being proposed and passed in the U.S. that would reshape animal welfare in industry and science. Bills such as Humane Slaughter Act, which was created to alleviate some of the suffering felt by livestock during slaughter, was passed in 1958. Later the Animal Welfare Act of 1966, passed by the 89th United States Congress and signed into law by President Lyndon B. Johnson, was designed to put much stricter regulations and supervisions on the handling of animals used in laboratory experimentation and exhibition but has since been amended and expanded. These groundbreaking laws foreshadowed and influenced the shifting attitudes toward nonhuman animals in their rights to humane treatment which Richard D. Ryder and Peter Singer would later popularize in the 1970s and 1980s.

Great ape personhood.

Great ape personhood is the idea that the attributes of nonhuman great apes are such that their sentience and personhood should be recognized by the law, rather than simply protecting them as a group under animal cruelty legislation. Awarding personhood to nonhuman primates would require that their individual interests be taken into account.

Speciesism () or specism denotes discrimination based on species membership. Such discrimination involves treating members of one species as morally more important than members of other species.

Speciesismus je druhová nadřazenost (obdoba rasismu, sexismu apod); diskriminace určitého živočišného druhu druhem jiným, popř. legitimizování jeho vykořisťování. Název je odvozen od latinského slova "species" = "druh", což je základní kategorie biologické nomenklatury. Termín je používán zejména ochránci práv zvířat, protože dobře vystihuje svoji podstatu.


History.

Origins.

Arguments among Jewish sects regarding the validity of the Oral Law date back to Hellenistic period, the 2nd and 1st centuries BCE. Accordingly, some scholars trace the origin of Karaism to those who rejected the Talmudic tradition as an innovation. Judah Halevi, an 11th-century Jewish philosopher and rabbi, wrote a defense for Judaism entitled "Kuzari", placing the origins of Karaism in the first and second centuries BCE, during the reign of Alexander Jannaeus ("King Jannai"), king of Judaea from 103 to 76 BCE:  Abraham Geiger, a 19th-century German scholar who founded Reform Judaism, posited a connection between the Karaites and a remnant of the Sadducees, the 1st-century Jewish sect that followed the Hebrew Bible literally and rejected the Pharisees' notion of an Oral Torah even before it was written. Geiger's view is based on comparison between Karaite and Sadducee "halakha": for example, a minority in Karaite Judaism

9th century.

Anan ben David (, c. 715 – 795 or 811?) is widely considered to be a major founder of the Karaite movement. His followers were called Ananites; they did not believe the rabbinical oral law was divinely inspired.  According to a 12th-century Rabbanite account, in approximately 760, Shelomoh ben Ḥisdai II, the Exilarch in Babylon died, and two brothers among his nearest kin, Anan ben David (whose name according to the Rabbanite account was Anan ben Shafaṭ, but was called "ben David" due to his Davidic lineage) and Ḥananyah were next in order of succession. Eventually Ḥananyah was elected by the rabbis of the Babylonian Jewish colleges (the Geonim) and by

The Golden Age.

In the "Golden Age of Karaism" (900–1100) a large number of Karaite works were produced in all parts of the Muslim world, the most notable being a work penned by Jacob Qirqisani, entitled "Kitāb al-Anwār wal-Marāqib" ("Code of Karaite Law"), which provides valuable information concerning the development of Karaism and throws light also on many questions in Rabbinic Judaism. Karaite Jews were able to obtain autonomy from Rabbanite Judaism in the Muslim world and

Anecdote about Karaites in Egypt during the 13th-century.

Egypt had long been a bastion for Karaites and their teachings. According to Rabbi David ibn Zimra, in one day in Egypt, a great

Karaites in the Russian Empire.

During the 19th century, Russian authorities began to differentiate Karaite Jews from Rabbanite Jews, freeing them from various oppressive laws that affected Rabbinic Jews. In the 1830s the Tsarist governor of the Taurida Governorate, Mikhail Semyonovich Vorontsov, told the Karaite leaders that, even though the Russian Empire liked the idea that the Karaites did not accept the Talmud, they were still Jews and responsible for the crucifixion of Jesus and thus subject to the laws. The leaders, hearing that, devised a ruse by which they could be freed of the oppressive laws and told him that

Six-Day War.

All Jewish men in Egypt were placed in camps during the Six-Day War, and they

Beliefs.

Karaite interpretations of the Torah.

Karaite Jews do not object to the idea of a body of interpretation of the Torah, along with extensions and development of non-rabbinic "halakha" (Jewish law) that strives to adhere to the Tanakh's straightforward meaning. Several hundred such books have been written by various Karaite Ḥakhamim (sages) throughout the movement's history, although most are lost today. The disagreement arises over the rabbinic tradition's raising of the Talmud and the other writings of the rabbis above the Torah. The Karaites believe this has led to traditions and customs being kept under rabbinic law that contradict what is written in the Torah. The Karaites also have their own traditions and customs passed down from their ancestors and religious authorities. These are known as "Sevel HaYerushah", which means "the Yoke [or burden] of Inheritance." Most of these practices are kept primarily by traditional Karaites; theoretically, any tradition thereof is said to be rejected if it

Shabbat.

As with other Jews, during Shabbat, Karaites attend synagogues to worship and to offer prayers. Most Karaites refrain from sexual relations on that day since they maintain that engaging in them can cause fatigue and copulation, in particular, results in ritual impurity on this holy day, concerns that Rabbinic Judaism ceased to have long ago; additionally, impregnating one's wife is considered "melakha" (forbidden work). Their prayer books are composed almost completely of biblical passages. Karaite Jews often practice full prostration during prayers, while most other Jews no longer pray in this fashion.  Unlike Rabbinic Jews, Karaites do not practice the ritual of lighting Shabbat candles. They have a differing interpretation of the Torah verse, "You shall not [burn] (Hebrew: "bi‘er" the "pi‘el" form of "ba‘ar") a fire in any of your dwellings on the day of Shabbat." In Rabbinic Judaism, the "qal" verb form "ba‘ar" is understood to mean "burn", whereas the "pi‘el" form (present here) is understood to be, not intensive as usual but

Laws of ritual purity.

Karaites maintain that in the absence of a Temple, ordinary washing with flowing waters (described in the Torah as "living"—flowing—water) should be substituted for purification with water that includes ashes obtained through the red heifer burning ritual. Karaites believe that this was the

Sephirath Ha‘Omer and Shavu‘oth.

The Karaite method of counting the days from the offering of the ‘Omer is different from the rabbinic method. The Karaites understand the term "morrow after the Sabbath" in Leviticus 23:15–16 to refer to the weekly Sabbath, whereas Rabbinic Judaism interprets it as referring to the day of rest on the

Tzitzit (Ṣiṣiyot).

A Tzitzit (alternatively spelled Ṣiṣit, plural: Tzitziyot or Ṣiṣiyot) is a knotted or braided tassel worn by observant Jews (both Karaite and Rabbinic) on each of the four corners of what is often an outer garment or their Tallit. The Torah commands Israel to make tassels on the corners of their four-cornered garment containing a thread of "Tekhelet" () and repeats this commandment using the word for "twisted cords" ("Gedilim") instead of "tassels" at. The purpose of the tassels is stated in the Book of Numbers as a visual reminder to the Israelites to remember the commandments given by God.  The thread of Tekhelet is a blue-violet or blue thread, which, according to the traditions of Rabbinic Judaism,

Tefillin.

Karaite Jews do not wear tefillin in any form. According to Karaites, the Biblical passages cited for this practice are metaphorical, and mean to "remember the Torah always and treasure it". This is because the commandment in scripture is "And these words, which I command thee this day, shall be upon thy heart"... "And thou shalt bind them for a sign upon thy hand, and they shall be for forehead ornaments between thine eyes". (Deuteronomy 6:5,9) Since words cannot be on

Mezuzot.

Like Tefillin, Karaites interpret the scripture that mandates inscribing the Law on doorposts and city gates as a metaphorical admonition, specifically, to keep the Law at home and away. This is because the previous commandment in the same passage is the source for Tefillin for Rabbinic Judaism, and is understood metaphorically due to the

Mamzerim.

In both Deuteronomy 23:3, and Zechariah 9:6, the Hebrew word "mamzer" is referenced alongside the nations of Ammon and Moab (in Deut 23:3), and the Philistine cities of Ashkelon, Gaza, Ekron and Ashdod (in Zech 9:5–6). From such, Karaites have come to consider the most logical understanding of the Hebrew word "mamzer",

Four species.

Karaite Judaism defines the Four species ("Arba`at haMinim") somewhat differently than Rabbinic Jews, i.e. (1) fruit of splendorous tree ("Peri `Eṣ Hadar"), which need not be the Etrog (yellow citrus fruit) demanded by rabbinic law, yet may be either any seasonal fruit tree considered splendorous by an individual Jew, or branches of olive trees featuring the olives, that were considered splendorous by the Judean Israelites in the generation of Nehemiah, as seen in Neḥemyah 8; (2) date palm fronds ("Kappoth Temarim") instead of the closed palm frond used by Rabbinic Jews;

Who is a Jew?

Karaite Judaism follows patrilineal descent, meaning a Jew is either someone whose father is Jewish (since almost all Jewish descent in the Tanakh is traced patrilineally) or both of whose parents are Jews, or one who has undergone a formal conversion which entails circumcision for uncircumcised males and formally accepting the God of Israel as one's own God and the people of Israel as one's own people.

Pronouncing the Tetragrammaton.

Some, such as the Karaites Jews in Khorasan (Persia) viewed it in Medieval times as a Mitzvah to do so because the name appears some 6800 times throughout the Tanakh (the Jewish Bible). Today, publicly uttering the Tetragrammaton (4 letter name) of God in Hebrew is a controversial issue among Karaites and indeed all Jews. Virtually all traditional Karaites

Writings.

Karaism has produced a vast library of commentaries and polemics, especially during its "Golden Age". These writings prompted new and complete defenses of the Mishnah and the Talmud, the culmination of these in the writings of Saadia Gaon and his criticisms of Karaism. Though he opposed Karaism, the rabbinic commentator Abraham ibn Ezra regularly quoted Karaite commentators, particularly Yefet ben Ali, to the degree that a legend exists among some Karaites that Ibn Ezra was ben Ali's student.  The most well-known Karaite polemic is Yiṣḥaq ben Avraham of Troki's "Faith Strengthened" (), a comprehensive counter-Christian

Karaites, Aharon Ben Mosheh Ben Asher, and the Masoretic Text.

Aaron ben Moses ben Asher was a Jewish scholar from Tiberias, famous as the most authoritative of the Tiberian Masoretes, and a member of a family who had been involved in creating and maintaining the Masoretic Text (authoritative text of the Hebrew scripture), for at least five generations. His Tiberian vocalization of the Bible is still, for all intents and purposes, the text all Jews continue to use, and he was the first systematic Hebrew grammarian.  His "Sefer Diqduqei HaTe‘amim" ("Grammar of the Punctuation/Vocalizations") was an original collection of grammatical rules and Masoretic information. Grammatical principles were not at that time considered worthy of independent study. The value of this work is that the grammatical rules presented by ben Asher reveal the linguistic background of vocalization for the first time. He

Karaites today.

Estimates of the size of the modern Karaite movement put the number at 1,500 Karaites in the United States, some 80 Constantinopolitan Karaites in Turkey, 30,000 in Israel, the largest communities being in Ramla, Ashdod and Beersheba, 1,196 in Ukraine (with Crimea), 300 in Lithuania and 205 in Russia. At the Polish census of 2002, only 45 people declared themselves "Karaims", including 43 Polish citizens.  During the early 1920s a British mandate official in Jerusalem records visiting the Karaite synagogue, which he describes as being "small, mediaeval, semi-underground" serving "Jerusalem's tiny colony of Qaraites".  In the early 1950s, the Chief Rabbinate of Israel originally objected to the immigration of Karaite Jews to Israel, and unsuccessfully tried to obstruct it. In 2007, however, Rabbi David Ḥayim Chelouche, the chief rabbi of Netayana was quoted in "The Jerusalem Post" as saying, "A Karaite is a Jew. We accept them as Jews and every one of them who wishes to come back [to mainstream Judaism] we accept back. There was once a question about whether Karaites needed to undergo a token circumcision in order to switch to Rabbinic Judaism, but the rabbinate agrees that today that is not necessary."  Moshe Marzouk, one of the Egyptian Jews executed in 1954 for planting bombs in Cairo in the service of Israeli Military Intelligence (the Lavon Affair) was a Karaite. Marzouk was considered a hero in Israel; however, his Karaite identity was downplayed in newspapers, which usually just described him as an Egyptian Jew. However, in 2001, the Israeli government, through the Israel Postal Authority, issued a special memorial sheet honoring

Karaism in Rabbinic Jewish opinion.

Rabbinic Judaism's scholars, such as Maimonides, write that people who deny the divine authority of the Oral Torah are to be considered among the heretics. However, at the same time, Maimonides holds ("Hilkhot Mamrim" 3:3) that most of the Karaites and others who claim to deny the "oral teachings" are not to be held accountable for their errors in the law because they are led into error by their parents and are similar to a tinok shenishba (a captive baby), or to one who was forced.  Rabbinic scholars have traditionally held that, because the Karaites do not observe the rabbinic law on divorce, there is a strong presumption that they are mamzerim (adulterine bastards), so that marriage with them is forbidden even if they return to Rabbinic Judaism. Some recent Ashkenazi Ḥaredi scholars have held that Karaites should be regarded as Gentiles in all respects, though this is not universally accepted. They hasten to add that this opinion is not intended to insult the Karaites, but only to give individual Karaites the option of integrating into mainstream Judaism by way of conversion. In contrast, in 1971, Ovadia Yosef, who was then the Chief Rabbi of the Sefaradim and ‘Edot HaMizraḥ of Israel, proclaimed that Egyptian Karaites are Jews and that it is permissible for Rabbinic Jews to marry with them.  In response to the position taken by the Karaites in regards to the authority of the Talmud, Orthodox Judaism counters first that the majority of the Oral Law codified in the Mishnah and Talmud are the legal rulings of the last Sanhedrin, a body of 71 elders that made up the highest court of jurisprudence in ancient Israel, and that not all of the Oral Law are literally "Laws given to Moses on Mount Sinai". The decisions made by this High Court must be upheld, per se the Law of Moses (Deuteronomy 17), this gives their legal rulings divine authority. Karaites reject the authority of this Sanhedrin that developed during the second Temple period partly because it was an admixture of different people and not just priests and levites as mandated by the Torah. The Sanhedrin also took legal authority away from the descendants of Zadok who served as priests in the Temple in Jerusalem. This is essentially the same view held by the Sadducees and Boethusians during the second Temple period. Secondly Rabbinic Judaism points to the innumerable examples of biblical commandments that are either too ambiguous or documented in such a concise fashion that proper adherence could not be enforced on a national scale without the further legislation provided by the Talmud. Karaites respond that the Torah itself states "this law I am commanding you is not too hard for you, neither is it far off. It is not in the heavens, to say, who shall ascend into the heavens and bring it to us, and cause us to understand it, that we do it?" indicating the Torah could easily be understood by the average Israelite. Examples cited in Rabbinic Judaism as laws requiring rabbinic explanation include:  Orthodox Judaism also notes that the Torah was never meant to be observed as a personal covenant between the individual Jew and God, but a national covenant wherein the Torah functions as the constitution of Israel as a whole. Orthodox Jews point out that the Torah could never be enforced as a national law, as it was during the time of Joshua, King David, and Ezra, if every individual Jew had their own opinion on how to observe its commandments. In order for the Torah to properly govern the Jewish people, and for its laws be legally enforced with the punishments and penalties prescribed in the Torah, those laws must be legislated and clearly defined by a ruling Sanhedrin. Karaites counter that the enforcement of the Torah on a national scale can only be legislated by the descendants of Zadok in the Temple in Jerusalem as per the Torah (Deuteronomy 17) and prophets (Ezekiel 44) not by a collection of opinions by various rabbis.  For Karaites, in sum, the rabbinic interpretations above, as codified in oral law, are only one form of interpretation. They are not divinely ordained, and they are neither binding halakhah nor practical religious law.

Rabbinical classification of a Jew.

A person whose mother was a Karaite Jew is regarded as halakhically Jewish by the Orthodox Rabbinate. By contrast, somebody who is patrilineally Jewish (one whose father is Jewish) is regarded as a Jew by the "Mo′eṣet HaḤakhamim" (the Karaite Council of Sages) on the condition that they were raised Jewish during childhood. Although it is widely accepted that Karaite Jews are halakhically Jewish (apparently with the exception of those who join the Jewish people through the Karaite movement), there is still a question as to whether or not marriage between

Karaite Judaism () or Karaism (; ; also spelt Qaraite Judaism or Qaraism) is a Jewish religious movement characterized by the recognition of the written Torah alone as its supreme authority in "halakha" (Jewish religious law) and theology. Karaites maintain that all of the divine commandments handed down to Moses by God were recorded in the written Torah without additional Oral Law or explanation. It is distinct from mainstream Rabbinic Judaism, which considers the Oral Torah, as codified in the Talmud and subsequent works, to be authoritative interpretations of the Torah. As a result, Karaite Jews do not accept as binding the written collections of the oral tradition in the Midrash or Talmud.

Karaité (hebrejsky קָרָאִים,בני מקרא,בעלי מקרא, "Kara'im", "Bnej Mikra", "Ba'alej Mikra"; arabsky "Qaraijjun", doslova: "Písmáci", "Synové Písma", "Vlastníci Písma") jsou stoupenci židovského hnutí, někdy označovaného za sektu, které existuje od 8. století. Doktrína karaitského judaismu je charakteristická odmítnutím ústní Tóry a uznáváním pouze Tanachu (Starý zákon) jako normativního zdroje veškerých právních a náboženských tradic, čímž se odlišuje od hlavního směru judaismu, který je označován jako rabínský judaismus. Název pochází z hebrejského kořene קרא – „číst“.