In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are called points; the metric satisfies a few simple properties. Informally: the distance from a point to itself is zero, the distance between two distinct points is positive, the distance from A to B is the same as the distance from B to A, the distance from A to B is less than or equal to the distance from A to B via any third point C. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces; the most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance; the Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

In 1906 Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel. However the name is due to Felix Hausdorff. A metric space is an ordered pair where M is a set and d is a metric on M, i.e. a function d: M × M → R such that for any x, y, z ∈ M, the following holds: Given the above three axioms, we have that d ≥ 0 for any x, y ∈ M. This is deduced as follows: The function d is called distance function or distance. D is omitted and one just writes M for a metric space if it is clear from the context what metric is used. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads; the triangle inequality expresses the fact. If the distance between two points is zero, the two points are indistinguishable from one-another. Many of the examples below can be seen as concrete versions of this general idea.

The real numbers with the distance function d = | y − x | given by the absolute difference, more Euclidean n-space with the Euclidean distance, are complete metric spaces. The rational numbers with the same distance function form a metric space, but not a complete one; the positive real numbers with distance function d = | log ⁡ | is a complete metric space. Any normed vector space is a metric space by defining d = ‖ y − x ‖, see metrics on vector spaces. Examples: The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points, or vectors, is the sum of the differences between corresponding coordinates; the maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from x to y. The British Rail metric on a normed vector space is given by d = ‖ x ‖ + ‖ y ‖ for distinct points x and y, d = 0. More ‖. ‖ can be replaced with a function f taking an arbitrary set S to non-negative reals and taking the value 0 at most once: the metric is defined on S by d = f + f for distinct points x and y, d = 0.

The name alludes to the tendency of railway journeys to proceed via London irrespective of their final destination. If {\displ

The Muyejebo is the oldest extant Korean martial arts manual, written during the reign of King Seonjo. The king died before the compendium was complete, it was first published, with the addition of material from Japanese martial arts, in 1610; as the Imjin War dragged on for years, Korea needed a way to and efficiently train a large number of troops, the Korean military adopted a training methodology based on a Ming dynasty Chinese military manual called the Jixiao Xinshu, written by the famed Chinese general, Qi Jiguang. The book was of particular interest to Koreans, as it was written by a Chinese commander who had defeated a major Japanese pirate force that had landed along the Southeast coast of China mere decades before the Imjin War began. Korean officials created their own version of the military training manual, based on the Chinese version, called it the Muyejebo. King Seonjo ordered his officials to add supplemental information to the Muyejebo, but died before the updated work was published.

His successor, King Gwanghaegun, continued the work of his father, which led to the publication of the Muyejebo sokjip by Choe Gi-nam. Around the time the book was to be published, four volumes of a Japanese martial arts manual were added as well, leading to the compilation of the Muyejebo Beonyeoksokjip in 1610. A woodcut edition of this updated manual still exists, is held by Keimyung University in Daegu, South Korea. In 1759, the book was once again published as the Muyesinbo. Both books, the Muyejebo and the Muyesinbo, formed the basis for the compilation of another, more famous Korean martial arts manual called the Muyedobotongji, published in 1791; the Muyejebo contains chapters about the use of the following weapons: Jangchang Ssangsudo Gonbang Deungpae wisteria shield and throwing spear Deungpae wisteria shield and waist sword Nangseon Dangpa The information regarding the use of the shield and throwing spear in combination with one another is part of the same chapter that covers the combined use of the shield and waist sword.

Only one copy of the original Muyejebo Beonyeoksokjip has survived to the present day. It was found in the Keimyung University library in Daegu, South Korea in 1998, was granted national treasure status in 2001; the Keimyung University Press republished the Muyejebo Beonyeoksokjip in 1999. Korean sword Korean martial arts The International hosinsool Federation

The Ulster Defence Association is an Ulster loyalist paramilitary group in Northern Ireland. It was formed in September 1971 as an umbrella group for various loyalist groups and undertook an armed campaign of twenty four years as one of the participants of the Troubles, its declared goal was to defend Ulster Protestant loyalist areas and to combat Irish republicanism the Provisional Irish Republican Army. In the 1970s, uniformed UDA members patrolled these areas armed with batons and held large marches and rallies. Within the UDA was a group tasked with launching paramilitary attacks; the British government outlawed the "UFF" in November 1973, but the UDA itself was not proscribed as a terrorist group until August 1992. The UDA/UFF were responsible for more than 400 deaths; the vast majority of its victims were Irish Catholic civilians, killed at random, in what the group called retaliation for IRA actions or attacks on Protestants. High-profile attacks carried out by the group include the Top of the Hill bar shooting, the Milltown massacre, the Sean Graham's and James Murray's bookmakers' shootings, the Castlerock killings, killings of Paddy Wilson and Irene Andrews and the Greysteel massacre.

Most of its attacks were in Northern Ireland, but from 1972 onward it carried out bombings in the Republic of Ireland. The UDA/UFF declared a ceasefire in 1994 and ended its campaign in 2007, but some of its members have continued to engage in violence; the other main Loyalist paramilitary group during the conflict was the Ulster Volunteer Force. All three groups are Proscribed Organisations in the United Kingdom under the Terrorism Act 2000; the Ulster Defence Association emerged from a series of meetings during the middle of 1971 of loyalist "vigilante" groups called "defence associations". The largest of these were the Shankill and Woodvale Defence Associations, with other groups based in East Belfast, the Hammer and Roden Street; the first meeting was chaired with Alan Moon as its vice-chair. Moon was replaced by Jim Anderson and had left the organisation by the time of its formal launch in September. By this point, Charles Harding Smith had become the group's leader, with former British soldier Davy Fogel as his second-in-command, who trained the new recruits in military tactics, the use of guns, unarmed combat.

Its most prominent early spokesperson was Tommy Herron. Its original motto was Cedenta Arma Togae and it was a legal organisation until it was banned by the British Government on 10 August 1992. At its peak of strength it held around forty thousand members part-time. During this period of legality, the UDA committed a large number of attacks using the name Ulster Freedom Fighters, including the murder of Social Democratic and Labour Party politician Paddy Wilson and his companion Irene Andrews in 1973; the UDA was involved in the successful Ulster Workers Council Strike in 1974, which brought down the Sunningdale Agreement – a power-sharing agreement for Northern Ireland, which some unionists thought conceded too much to nationalist demands. The UDA enforced this general strike through widespread intimidation across Northern Ireland; the strike was led by UDA member, Glenn Barr. The UDA were referred to by the nickname "Wombles" by their rivals the Ulster Volunteer Force; the nickname is derived from the furry fictional children's TV creatures The Wombles, was given to the UDA because many of its members wore fur-trimmed parkas.

Its headquarters is in Gawn Street, off the Newtownards Road in east Belfast, its current motto is Quis Separabit, Latin for "Who will separate?" The UDA had several women's units. Although they helped staff roadblocks, the women's units were involved in local community work and responsible for the assembly and delivery of food parcels to UDA prisoners; this was a source of pride for the UDA. The first women's unit was founded on the Shankill Road by Wendy "Bucket" Millar, whose sons Herbie and James "Sham" Millar would become prominent UDA members; the UDA women's department was headed by Jean Moore, who came from the Shankill Road. She had served as the president of the women's auxiliary of the Loyalist Association of Workers, her brother Ingram "Jock" Beckett, one of the UDA's founding members, had been killed in March 1972 by a rival UDA faction in an internal dispute. Moore was succeeded by Hester Dunn of east Belfast, who ran the public relations and administration section at the UDA headquarters.

Wendy Millar's Shankill Road group was a active women's unit, another was based in Sandy Row, south Belfast, a traditional UDA stronghold. The latter was commanded by Elizabeth "Lily" Douglas, her teenaged daughter, Elizabeth was one of the members. The Sandy Row women's UDA unit was disbanded after it carried out a vicious "romper room" punishment beating on 24 July 1974 which left 32-year-old Ann Ogilby dead; the body of Ogilby, a Protestant single mother who had an affair with the husband of one of the unit's members, was found in a ditch five days later. The day of the fatal beating Ogilby was abducted and forced upstairs to the first floor of a disused bakery in Sandy Row, converted into a UDA club. Two teenage girls, Henrietta Cowan and Christine Smith, acting under Elizabeth Douglas' orders to give Ogilby a "good rompering", kicked battered her to death with bricks and sticks; the killing, carried out within earshot of