hotchpotch/wikipedia-multilingual-ir-related-lead-pairs huggingface dataset:
Modes of convergence (annotated index)
The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships.  For an expository article, see Modes of convergence.  Simple logical relationships between different modes of convergence are indicated (e.g., if one implies another), formulaically rather than in prose for quick reference, and indepth descriptions and discussions are reserved for their respective articles.

Guide to this index.  To avoid excessive verbiage, note that each of the following types of objects is a special case of types preceding it: sets, topological spaces, uniform spaces, topological abelian groups (TAG), normed vector spaces, Euclidean spaces, and the real/complex numbers.  Also note that any metric space is a uniform space.  Finally,  subheadings will always indicate special cases of their super headings.

The following is a list of modes of convergence for:

Uniformly Cauchy sequence
In mathematics, a sequence of functions  from a set S to a metric space M is said to be uniformly Cauchy if:

For all , there exists  such that for all :  whenever .

Another way of saying this is that  as , where the uniform distance  between two functions is defined by
Cairate
Cairate is a comune (municipality) in the Province of Varese in the Italian region Lombardy, located about 35 kilometres (22 mi) northwest of Milan and about 15 kilometres (9 mi) south of Varese.

Cairate borders the following municipalities: Carnago, Cassano Magnago, Castelseprio, Fagnano Olona, Locate Varesino, Lonate Ceppino, Tradate.

Lonate Ceppino
Lonate Ceppino is a comune (municipality) in the Province of Varese in the Italian region Lombardy, located about 35 kilometres (22 mi) northwest of Milan and about 13 kilometres (8 mi) south of Varese.

Lonate Ceppino borders the following municipalities: Cairate, Castelseprio, Gornate-Olona, Tradate, Venegono Inferiore. It is crossed by the Olona and its affluent, the Bozzone.

The name of the town has Celtic origins. The word Lonate seems to derive from the Celtic word Lona, which means water puddle. The name's origin is also confirmed by the presence of a water spring, which is located near the old church. In ancient times, this water spring was believed to be miraculous.
Sangtam language
Sangtam, also called Thukumi, Isachanure, or Lophomi, is a Naga language spoken in northeast India. It is spoken in Kiphire District and in the Longkhim-Chare circle in Tuensang district, Nagaland, India.

Naga languages
The Naga languages are a geographic and ethnic grouping of Tibeto-Burman, spoken mostly by Naga peoples.

Northern Naga languages do not fall within the group, in spite of being spoken by Naga groups; instead, these form part of the Sal languages within Sino-Tibetan, while Southern Naga languages form a branch within Kuki-Chin languages subfamily.
Wolf Hirth
Wolfram Kurt Erhard Hirth (28 February 1900 – 25 July 1959) was a German gliding pioneer and sailplane designer. He was a co-founder of Schempp-Hirth, still a renowned glider manufacturer.

Hirth was born in Stuttgart, the son of an engineer and tool-maker. He was the younger brother of Hellmuth, who founded the famous Hirth aircraft engine manufacturing company.

Hellmuth Hirth
Hellmuth Hirth (German: [ˈhɛlmuːt ˈhɪʁt] ⓘ; 24 April 1886 – 1 July 1938) was a German engineer who founded the Mahle GmbH and Hirth companies, manufacturing engine components and complete aircraft engines respectively.
Ramsden Heath
Ramsden Heath is a village in Essex in the east of England. It is located approximately 8 mi (13 km) south of the county town of Chelmsford; the closest towns are Billericay, approximately 3 mi (4.8 km) west-south-west, and  Wickford, approximately 3.5 mi (5.6 km) south-east. The village is close to Hanningfield Reservoir and the smaller village of Downham lies immediately to the east.

In 2018, Ramsden Heath had an estimated population of 1,966  and has approximately 600 households. The parish council is South Hanningfield, the local authority is the City of Chelmsford and it is sited within the parliamentary constituency of Maldon.

Downham, Essex
Downham is a small village in the civil parish of South Hanningfield, in the Chelmsford district of Essex, England. It is located approximately 7 miles (11 km) south of the county town of Chelmsford. The nearest towns are Billericay, 3+1⁄2 miles (5.6 km) west-south-west, and Wickford, 2+1⁄4 miles (3.6 km) south-east.
Želeče
Želeče (pronounced [ʒɛˈleːtʃɛ], formerly Bled–Želeče; German: Schalkendorf) is a former settlement in the Municipality of Bled in northwestern Slovenia. It is now part of the town of Bled. The area is part of the traditional region of Upper Carniola and is now included with the rest of the municipality in the Upper Carniola Statistical Region.

Bled
Bled (pronounced [ˈbléːt] ⓘ; German: Veldes, in older sources also Feldes) is a town on Lake Bled in the Upper Carniolan region of northwestern Slovenia. It is one of Slovenia's most visited tourist destinations. The town is the administrative seat of the Municipality of Bled.
Der Morgenstern ist aufgedrungen
"Der Morgenstern ist aufgedrungen" (The morning star is risen) is an Advent song and Christmas carol with lyrics by Lutheran minister Daniel Rumpius (or Rump), published first in 1587. The common melody by Michael Praetorius appeared later in 1609. It was reprinted, slightly revised by Otto Riethmüller in 1932, and is part of the current Protestant hymnal Evangelisches Gesangbuch and other song books.

Advent song
Advent songs (German: Adventslieder) are songs and hymns intended for Advent, the four weeks of preparation for Christmas. Topics of the time of expectation are the hope for a Messiah, prophecies, and the symbolism of light, among others. Several of the songs are part of hymnals such as the German Catholic Gotteslob (GL) and the Protestant Evangelisches Gesangbuch (EG).

This is a dynamic list and may never be able to satisfy particular standards for completeness.
Bremen Hut (Ilsenburg)
The Bremen Hut (German: Bremer Hütte) in the Harz Mountains is a refuge hut and shelter in that part of the Harz National Park lying within the borough of Ilsenburg (Harz) in Harz district in the German state of Saxony-Anhalt.

Ilsenburg
Ilsenburg (German: [ˈɪlzn̩bʊʁk] ⓘ) is a town in the district of Harz, in Saxony-Anhalt in Germany. It is situated under the north foot of the Harz Mountains, at the entrance to the Ilse valley with its little river, the Ilse, a tributary of the Oker, about six 6 miles (9.7 km) north-west of the town of Wernigerode. It received town privileges in 1959. Owing to its surrounding of forests and mountains as well as its position on the edge of the Harz National Park, Ilsenburg is a popular tourist resort. Since 2002, it is officially an air spa.
U.S. Route 395 in California
U.S. Route 395 (US 395) is a United States Numbered Highway, stretching from Hesperia, California to the Canadian border in Laurier, Washington. The California portion of US 395 is a 557-mile (896 km) route which traverses from Interstate 15 (I-15) in Hesperia, north to the Oregon state line in Modoc County near Goose Lake. The route clips into Nevada, serving the cities Carson City and Reno, before returning to California.

Prior to truncation, US 395 served the metropolitan areas of San Diego, Riverside, and San Bernardino. The highway serves as a connection to the Greater Los Angeles area for the communities of the Owens Valley, Mammoth Lakes and Mono Lake. The highway is used as an access route for both the highest point in the contiguous United States, Mount Whitney, and the lowest point in North America, Death Valley.

The corridor has been used since the California gold rush, and before numbering was known by several names including El Camino Sierra.

Lone Pine, California
Lone Pine is a census-designated place (CDP) in Inyo County, California, United States, located 16 mi (26 km) south-southeast of Independence. The population was 2,014 at the 2020 census, down from 2,035 at the 2010 census. The town is located in the Owens Valley, near the Alabama Hills and Mount Whitney, between the eastern peaks of the Sierra Nevada to the west and the Inyo Mountains to the east. The local hospital, Southern Inyo Hospital, offers standby emergency services. The town is named after a solitary pine tree that once existed at the mouth of Lone Pine Canyon. On March 26, 1872, the very large Lone Pine earthquake destroyed most of the town and killed 27 of its 250 to 300 residents.
Lucille Caudill Little
Lucille Caudill Little (August 20, 1909 – October 8, 2002) was an American patron of the arts and philanthropist who served as president of the W. Paul and Lucille Caudill Little Foundation in Lexington, Kentucky.

Morehead State University
Morehead State University (MSU) is a public university in Morehead, Kentucky, United States. The university began as Morehead Normal School, which opened its doors in 1887. The Craft Academy for Excellence in Science and Mathematics, a two-year residential early college high school on the university's campus, was established in 2014.
Yongsheng County
Yongsheng County (simplified Chinese: 永胜县; traditional Chinese: 永勝縣; pinyin: Yǒngshèng Xiàn) is located in the northwest of Yunnan province, China. It is under the administration of the prefecture-level city of Lijiang. In 2019 the county had a population of 406,757 including 34.42% ethnic minorities.

The Chenghai Lake is located in Yongsheng. Yongsheng has a strong agricultural output consisting especially of fruits including pomegranate, oranges, grapes, mangoes, longan,  and Sugar-apple.

Yongsheng County borders Huaping County to the east, Dayao County to the southeast, Yulong County and Heqing County to the west, Ninglang County and Gucheng District to the north and Binchuan County to the south.

Ninglang Yi Autonomous County
Ninglang Yi Autonomous County (simplified Chinese: 宁蒗彝族自治县; traditional Chinese: 寧蒗彝族自治縣; pinyin: Nínglàng Yízú Zìzhìxiàn; Yi: ꆀꆿꆈꌠꊨꏦꏱꅉꑤ, nip lat nuo su zyt jie jux dde xiep) is located in the northwest of Yunnan province, China, bordering Sichuan province to the north, east and northeast. It is under the administration of Lijiang City. The county is home to the Mosuo people, who lived under the quasi-independent Chiefdom of Yongning until abolished in 1956. Ninglang Luguhu Airport is located in the county.
Daniel Bellissimo
Daniel Bellissimo (born August 15, 1984) is a Canadian-born Italian retired professional ice hockey goaltender. He last played with Luleå HF of the Swedish Elitserien. Is active GM of Streetsville in the SOHL.

Born in Toronto, Bellissimo played junior hockey in the OPJHL for the Vaughan Vipers. He moved to the Italian team HC Asiago in 2006, where he played with his brother Vince. He won two Top League title in Italy.

Thanks to his Italian origins, he participated at the 2010 IIHF World Championship as a member of the Italy men's national ice hockey team.

Vince Bellissimo
Vince Bellissimo (born December 14, 1982) is a Canadian former professional ice hockey player. Bellissimo was selected by the Florida Panthers in the 5th round (158th overall) of the 2002 NHL Entry Draft.
New Year's Eve
New Year's Eve in the Gregorian calendar refers to the evening—or commonly the entire day—of the last day of the year: 31 December. In many countries, New Year's Eve is celebrated with dancing, eating, drinking, and watching or lighting fireworks. Many Christians attend a watchnight service to mark the occasion. New Year's Eve celebrations generally continue into New Year's Day, 1 January, past midnight.

The local time zone determines the advent of the New Year; the first places to welcome the New Year are west of the International Date Line: the Line Islands (part of Kiribati), Samoa and Tonga, in the Pacific Ocean. In contrast, American Samoa, Baker Island and Howland Island (part of the United States Minor Outlying Islands) are among the last.

Molybdomancy
Molybdomancy (from Ancient Greek: μόλυβδος, romanized: molybdos, lit. 'lead' and -mancy) is a technique of divination using molten metal. Typically, molten lead or tin is dropped into water. It can be found as a tradition in various cultures, including Austria, Bosnia and Herzegovina, Bulgaria, Germany, Finland, Estonia, Latvia, Switzerland, the Czech Republic, and Turkey. Some versions have been found to have potentially harmful effects on human health.
Stefano Scaini
Stefano Scaini (born 4 May 1983) is a former Italian male long-distance runner who competed at three editions of the IAAF World Cross Country Championships at senior level (2003, 2005, 2006). He is the husband of the marathon runner Anna Incerti.

Anna Incerti
Anna Carmela Incerti (born 19 January 1980 in Palermo) is an Italian long-distance runner who specializes in the marathon. She has represented Italy in the marathon at European, World and Olympic-level. She won the bronze in the event at the 2010 European Championships, later upgraded to silver and then to gold.
Ap Lei Chau Estate
Ap Lei Chau Estate (Chinese: 鴨脷洲邨) is a public housing estate in Ap Lei Chau, Hong Kong Island, Hong Kong. It is the first public housing estate in Ap Lei Chau. Completed in two phases in 1980 to 1982, respectively, the estate consists of 8 residential blocks providing 4,453 flats. It was one of the public housing estates built from 1980 to 1982 to accommodate people affected by a major fire in Aberdeen Typhoon Shelter.

Ap Lei Chau
Ap Lei Chau or Aberdeen Island is an island of Hong Kong, located off Hong Kong Island next to Aberdeen Harbour and Aberdeen Channel. It has an area of 1.30 square kilometres (0.50 sq mi) after land reclamation. Administratively it is part of the Southern District. Ap Lei Chau is one of the most densely populated islands on earth.

In the 2000s, the Guinness World Records called it the world's most densely populated island on their website.
Eduard Grikurov
Eduard Grikurov  (29 March 1907, Tbilisi - 13 December 1982, Leningrad) was a Soviet conductor and People's Artist of the USSR.

Grikurov studied at the Conservatory in Tbilisi with Mikhail Ippolitov-Ivanov, and worked regularly with the Moscow Philharmonic Orchestra, the Kirov Opera (today Mariinsky Theatre), and the Leningrad Philharmonic Orchestra. From 1944 until 1969 Grikurov served as music director of the Leningrad State Academic Maly Opera Theatre (today Mikhailovsky Theater).

He was awarded the Stalin Prize in 1951, and was named People's Artist of Russia in 1957.

Until his death he served as professor of conducting at the Leningrad Conservatory. His most notable students are Alexander Alexeev, Vakhtang Jordania, and Dmitri Kitayenko.

Mikhail Ippolitov-Ivanov
Mikhail Mikhailovich Ippolitov-Ivanov (Russian: Михаи́л Миха́йлович Ипполи́тов-Ива́нов; born Mikhail Mikhailovich Ivanov; 19 November [O.S. 7 November] 1859 – 28 January 1935) was a Russian and Soviet composer, conductor and teacher. His music ranged from the late-Romantic era into the 20th century era.
Dianthus giganteus
Dianthus giganteus, the giant pink, is a species of pink native to Romania, the Balkan peninsula, and possibly nearby areas. A perennial with flowerheads reaching 1 m, it is useful in gardening and landscaping applications where a backdrop of taller plants is needed. It is available from commercial suppliers, with the Royal Horticultural Society considering it to be a good plant to attract pollinators.

Dianthus
Dianthus (/daɪˈænθəs/ dy-AN-thəs) is a genus of about 340 species of flowering plants in the family Caryophyllaceae, native mainly to Europe and Asia, with a few species in north Africa and in southern Africa, and one species (D. repens) in arctic North America. Common names include carnation (D. caryophyllus), pink (D. plumarius and related species), and sweet william (D. barbatus).
Byron, Minnesota
There are also two Byron Townships in Minnesota.

Byron is a city in Olmsted County, Minnesota, United States. About 10 miles west of the city of Rochester and 6 miles east of the city of Kasson on U.S. Route 14. It is surrounded by Kalmar Township. The population was 6,312 at the 2020 census, and was estimated at 6,809 in 2024. Byron is part of the Rochester metropolitan area.

Local industries are in the form of farm services and printing. A grain elevator is situated next to the rail line that runs through town, which is owned by the Dakota, Minnesota and Eastern Railroad. Schmidt Printing (a subsidiary of Taylor Corporation) is another major company in town. Byron is also a bedroom community for nearby Rochester, Minnesota.

Rochester metropolitan area, Minnesota
The Rochester metropolitan statistical area, as defined by the United States Census Bureau, is a metropolitan statistical area consisting of four counties in southeast Minnesota, Dodge, Fillmore, Olmsted, and Wabasha. The principal city is Rochester, the city, which has a population of 121,395.

A July 1, 2024 census estimate placed the population at 230,742, an increase of 1.95% from the 2020 census, an increase of 11.54% from the 2010 census, and an increase of 24.90% from the 2000 census.
Trymedia
Trymedia Systems, Inc. is a division of RealNetworks that provides digital distribution services based on its proprietary ActiveMARK DRM and digital distribution technology. Trymedia is headquartered in San Francisco, with offices in Berkshire and Alicante.

RealNetworks
RealNetworks LLC is an American technology company and provider of Internet streaming media delivery software and services based in Seattle, Washington. The company also provides subscription-based online entertainment services and mobile entertainment and messaging services.
Rothera Research Station
The Rothera Research Station is a British Antarctic Survey (BAS) base on the Antarctic Peninsula, located at Rothera Point, Adelaide Island. Rothera also serves as the capital of the British Antarctic Territory, a British Overseas Territory.

De Havilland Canada Dash 7
The de Havilland Canada DHC-7, popularly known as the Dash 7, is a turboprop-powered regional airliner with short take-off and landing (STOL) performance. Variants were built with 50–54 seats. It first flew in 1975 and remained in production until 1988 when the parent company, de Havilland Canada, was purchased by Boeing in 1986 and later sold to Bombardier. In 2006 Bombardier sold the type certificate for the aircraft design to Viking Air, which was later reorganized as De Havilland Canada.
Natural regions of Germany
This division of Germany into major natural regions takes account primarily of geomorphological, geological, hydrological, and pedological criteria in order to divide the country into large, physical units with a common geographical basis. Political boundaries play no part in this, apart from defining the national border.

In addition to a division of Germany by natural regions, the federal authorities have also produced a division by so-called landscape areas (Landschaftsräume) that is based more on human utilisation of various regions and so has clearly different boundaries.

Eifel
The Eifel (German: [ˈaɪfl̩] ⓘ; Luxembourgish: Äifel, pronounced [ˈæːɪ̯fəl]) is a low mountain range in western Germany, eastern Belgium and northern Luxembourg. It occupies parts of southwestern North Rhine-Westphalia, northwestern Rhineland-Palatinate and the southern area of the German-speaking Community of Belgium.

The Eifel is part of the Rhenish Massif; within its northern portions lies the Eifel National Park. The Eifelian stage in geological history is named after the region because rocks of that period reach the surface in the Eifel at the Wetteldorf Richtschnitt outcrop.

The inhabitants of the Eifel are known as Eiflers or Eifelers.
Paraneuretus
Paraneuretus is an extinct genus of formicid in the ant subfamily Aneuretinae known from fossils found in Asia and Europe.  The genus contains three middle to late Eocene age species, Paraneuretus dubovikoffi, Paraneuretus longicornis, and Paraneuretus tornquisti.

Aneuretinae
Aneuretinae is a subfamily of ants consisting of a single extant species, Aneuretus simoni (Sri Lankan relict ant), and 9 fossil species. Earlier, the phylogenetic position of A. simoni was thought to be intermediate between primitive and advanced subfamilies of ants, but recent studies have shown it is the nearest living relative of subfamily Dolichoderinae.
Pentucket Regional High School
Pentucket Regional High School (often abbreviated as "PRHS") is a public high school in West Newbury, Massachusetts, United States, which serves the communities of Groveland, Merrimac, and West Newbury. It is the only high school in the Pentucket Regional School District. As of 2023, the new school building was completed. The school's principal is Brenda Erhardt, also as of 2023.

Groveland, Massachusetts
Groveland is a town in Essex County, Massachusetts, United States. It is thirty-four miles north of Boston. At the 2020 census, it had a population of 6,752. The town is divided into two precincts, Groveland and South Groveland.
Abby Martin
Abby Martin is an American journalist, television presenter, and activist. She helped found the citizen journalism website Media Roots and serves on the board of directors for the Media Freedom Foundation which manages Project Censored. Martin appeared in the documentary film Project Censored The Movie: Ending the Reign of Junk Food News (2013), and co-directed 99%: The Occupy Wall Street Collaborative Film (2013).

She hosted Breaking the Set on the Russian state-funded network RT America from 2012 to 2015 and gained attention for condemning the Russian annexation of Crimea on-air, and then launched The Empire Files in that same year as an investigative documentary and interview series on Telesur, later released as a web series. In 2019, she released the film documentary The Empire Files: Gaza Fights for Freedom.

Michael Prysner
Michael D. Prysner (born June 15, 1983) is an American socialist and political activist. He is a U.S. Army veteran who served in Iraq as a specialist. His duties in Iraq included ground surveillance, home raids, and interrogation of prisoners. According to Prysner, these experiences led him to take an anti-war stance.
Googlebot
Googlebot is the web crawler software used by Google that collects documents from the web to build a searchable index for the Google Search engine. This name is actually used to refer to two different types of web crawlers: a desktop crawler (to simulate desktop users) and a mobile crawler (to simulate a mobile user).

Google Search Console
Google Search Console (formerly Google Webmaster Tools) is a web service by Google which allows webmasters to check indexing status, search queries, crawling errors and optimize visibility of their websites.

Until 20 May 2015, the service was called Google Webmaster Tools. In January 2018, Google introduced a new version of the search console, with changes to the user interface. In September 2019, old Search Console reports, including the home and dashboard pages, were removed.

Since 2019, Search Console has supported Domain properties, which combine all protocol and subdomain variations (such as http, https, www, and non-www) into a single property, simplifying site management and reporting.