Before giving a mathematically precise definition, we give a brief example.

Before giving the abstract definition we give a few examples:

Before Terry, every police encounter constituted an arrest, giving the police officer the full range of search authority.

Before reclamation and development, the low-lying areas around this hill were largely marsh or tidal wetland, giving the hill much more the appearance of a " ness " than today.

Before Charles's restoration, the Navigation Acts of 1650 had hurt Dutch trade by giving English vessels a monopoly, and had started the First Dutch War ( 1652 – 1654 ).

Before Richard is sent to his death, he " un-kings " himself by giving away his crown, sceptre, and the balm that is used to anoint a king to the throne.

Before giving birth ( usually in May ), the female will excavate a den to house her litter of four to seven kits.

Before giving it to her, Mama adds a healthy amount of cream.

Before the introduction of stamps, it was the recipient of mail — not the sender — who generally paid the cost of postage, giving the fee directly to the postman on delivery.

Before Egill died he allegedly concealed his silver treasure near Mosfellsbær, giving birth to the legend of silfur Egils (" Egill's Silver ").

Before the First Sino Japanese War, China engaged in technological modernization only, buying modern weapons, ships, artillery, and building modern arsenals to produce these weapons, and only giving their soldiers modern weapons without institutional reform, and while refusing to reform the government or civil society according to western standards, unlike Japan, who adopted western style government with a Parliament and completely reoganized the army on western lines.

Before giving its his blessings he set out the goals of such a pilgrimage.

Before definitely giving its consent to peace negotiations, the Roman Church determined on making a last effort to reduce the Hussites to subjection ; this resulted in the 5th Crusade against the Hussites.

Before she departed the show, Vance was offered a new contract with Desilu Studios, giving her the opportunity to direct.

Before giving his doctrinal exposition of a passage, he first considers its critical state.

Before Malpaso Productions co-founder Irving Leonard died, he and Eastwood discussed a final film, one giving Eastwood the artistic control he desired by making his directorial debut.

Before the election, the Labour Party ( in government ) held 56 seats, giving it an absolute majority in Parliament.

Before his short-lived comeback season in 2011, Forsberg never had a negative plus-minus rating, giving him an overall career rating of + 238.

Before giving lectures at St. George's Hospital, London, in 1895, Livingstone College afforded Manson his first opportunity to teach courses in tropical medicine.

Before the 1902 production of Twirly-Whirly, John Stromberg, who had composed several hit songs for her, delayed giving Lillian Russell her solo for several days, saying that it was not ready.

Before 1812, while Illinois had a first-grade territorial status, Edwards had vast powers to appoint county and local officials ; however, he made it his practice to consider local opinion as much as he could when making appointments, often giving weight to petitions signed by local residents.

Before Cogan was able to publish a second edition, two printers pirated the work, giving the author as " A Gentleman " rather than Hoyle.

Before the album was completed, Patty Valentine had brought a defamation-of-character suit against Dylan, regarding the song " Hurricane " from Desire ; on May 22, while giving a pre-trial deposition in his defense, Dylan was asked about his wealth.

Before leaving the court, Louis was thus instructed by his father: " In sending you to command my army, I am giving you an opportunity to make known your merit ; go and show it to all Europe, so that when I come to die it will not be noticed that the King is dead.

Before the 1890s, there was no formal party system in New South Wales.

Before the 1890s, there was no formal party system in Victoria.

Before the 1890s, there was no formal party system in Tasmania.

Before the 1890s when there was no formal party system in South Australia, MPs tended to have historical liberal or conservative beliefs.

Before the formal charter signing in 2000, URI supporters around the world participated together in a project called " 72 Hours for Peace ", in which more than 250 local organizations united in projects promoting peace and justice during the turn of the millennium.

Before receiving the formal nod, Harding was summoned by George Harvey, told he was considered to be the consensus nominee, and asked if he knew, " before God ", whether there was anything in his life which would be an impediment.

Before she began her education under Sophia Charlotte's care, Caroline had received little formal education ; her handwriting remained poor throughout her life.

Before being granted access to the library, new readers are required to agree to a formal declaration.

Before St. Mary ’ s was recognized as a senior college in 1925, there was no formal conference competition, so the rivalry between the downtown and Woodlawn campuses was fierce.

Before ABC Sports even became a formal division of the network, Scherick and ABC programming chief Tom Moore pulled off many programming deals involving the most popular American sporting events.

Before the new Constitution came into force, full formal powers to appoint the Prime Minister and the rest of the Council of State had been the privilege of the President, who was free to diverge from parliamentary principles, although ministers appointed had to have the confidence of the Parliament.

Before the Boundary Commission began formal hearings, governments were set up for the East and the West Punjab regions.

Before leaving Göttingen, Ernest penned a formal letter of thanks to the university, and wrote to his father, " I should be one of the most ungrateful of men if ever I was forgetful of all I owe to Göttingen & its professors.

Before this point there had been no formal demarcation between primary and secondary education as known in modern society.

Before European colonization, aboriginal warfare tended to be formal and ritualistic, and entailed relatively few casualties.

Before enacting the Goseibai Shikimoku, the Kamakura shogunate conducted trials without formal laws.

Before the formal declaration, Korea also aired the following complaints to be heard by the Japanese people through papers and media:

Before he stepped down as Harvard's president in 1933 and with Nazi rumblings overseas in Germany, Abbott Lawrence Lowell wished to break the stronghold of the German Ph. D. degree on American academic life, believing it stifled creativity with its overlong list of formal requirements.

Before widespread formal education, Cantonese opera taught morals and messages to its audiences rather than being solely entertainment.

Before the new constitution was enacted, President Boris Yeltsin had dissolved the Congress of People's Deputies and Supreme Soviet of Russia during the Russian constitutional crisis of 1993, although he did not have the formal constitutional powers to do so.

Before formal recognition as a modern breed in 1943, the type had been kept separate from other dogs as an undocumented breed for centuries in the rural province of Drenthe.

Before their formal designation as state reptiles, Florida's alligator, Maryland's terrapin, and Texas's horned lizard were all mascots of a major in-state university.

Before the formal orchestral ritornello leading into the recapitulation, Mozart writes a series of descending sequences with the cellos and bassoons holding suspensions over staccato strings.

In the period immediately before the Last Glacial Maximum, many areas that became completely barren desert were wetter than they are today, notably in southern Australia where Aboriginal occupation is believed to coincide with a wet period between 40, 000 and 60, 000 years BP ( Before Present, a formal measurement of uncalibrated radiocarbon years, counted from 1950 CE ).