In mathematics, a cover of a set is a collection of sets whose union contains as a subset. Formally, if
is an indexed family of sets , then is a cover of if
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 Uα of X whose union is the whole space X. In this case we say that C covers X, or that the sets Uαcover 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 open cover if each of its members is an open set (i.e. each Uα 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 which intersects only finitely many sets in the cover. Formally, C = {Uα} is locally finite if for any x ∈ X, there exists some neighborhood N(x) of x such that the set
Cover is the fourth solo album by Tom Verlaine released in 1984.
All songs written and composed by Tom Verlaine; except "Five Miles of You" composed with Jimmy Ripp
A cover system is video game gameplay mechanic that allows a virtual avatar to avoid dangers, usually in a three-dimensional world. This method is a digital adaptation of the real-life military tactic of taking cover behind obstacles, for purposes of attaining protection from enemy ranged or area effect attacks, such as gunfire or explosions. Similar gameplay elements can be traced back to as early as 1986, in Rolling Thunder. Later games which refined the system include Bonanza Bros., Blackthorne, Time Crisis, Metal Gear Solid, WinBack, Police 911, Splinter Cell, Kill Switch, Gears of War, Uncharted, Mass Effect and Vanquish.
In gaming, a cover system lets a player character use stationary or moving obstacles to avoid damage. To be considered a cover system, there must be some physical interaction with the source of cover and the avatar. This means standing behind an object, as in traditional shooter games, while strictly speaking would be classified as taking cover, does not qualify as an actual cover system. Some first-person shooters such as Soldier of Fortune bridged the gap somewhat by allowing players to lean to the sides, allowing the avatar to lean out from behind objects to survey the environment or open fire on the enemy, without fully moving their own bodies into the open. In addition, the player character must have the ability to move in and out of the covering objects' proximity, leaving points of vulnerability to the player. This excludes the exclusive use of portable shields as a cover system, though they may often be used to supplement a stationary source of cover, as seen in video games like Army of Two, and Gears of War 2.
To go "undercover" is to avoid detection by the entity one is observing, and especially to disguise one's own identity or use an assumed identity for the purposes of gaining the trust of an individual or organization to learn or confirm confidential information or to gain the trust of targeted individuals in order to gather information or evidence. Traditionally, it is a technique employed by law enforcement agencies or private investigators, and a person who works in such a role is commonly referred to as an undercover agent.
Undercover work has been used in a variety of ways throughout the course of history, but the first organized, but informal, undercover program was first employed in France by Eugène François Vidocq in the early 19th century. At the end of 1811, Vidocq set up an informal plainclothes unit, the Brigade de la Sûreté ("Security Brigade"), which was later converted to a security police unit under the Prefecture of Police. The Sûreté initially had eight, then twelve, and, in 1823, twenty employees. One year later, it expanded again, to 28 secret agents. In addition, there were eight people who worked secretly for the Sûreté, but instead of a salary, they received licences for gambling halls. A major portion of Vidocq's subordinates were ex-criminals like himself.
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.
Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.
The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.
Topology is an indie classical quintet from Australia, formed in 1997. A leading Australian new music ensemble, they perform throughout Australia and abroad and have to date released 12 albums, including one with rock/electronica band Full Fathom Five and one with contemporary ensemble Loops. They were formerly the resident ensemble at the University of Western Sydney. The group works with composers including Tim Brady in Canada, Andrew Poppy, Michael Nyman, and Jeremy Peyton Jones in the UK, and Terry Riley, Steve Reich, Philip Glass, Carl Stone, and Paul Dresher in the US, as well as many Australian composers.
In 2009, Topology won the Outstanding Contribution by an Organisation award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.
Bernard studied viola at the Queensland Conservatorium (B.Mus 1987) and at Michigan State University (Master of Music 1993) with John Graham and Robert Dan. He studied in summer schools with Kim Kashkashian (Aldeborough), the Alban Berg Quartet and the Kronos Quartet. While in the US, he played with the Arlington Quartet, touring the US and UK. He was a violist in the Queensland Philharmonic Orchestra from 1994-2000, and is now Associate Principal Violist of the Queensland Orchestra, playing solo parts in works such as the sixth Brandenburg Concerto. He has directed several concerts for the Queensland Philharmonic’s Off the Factory Floor chamber series.
In mathematics, a cover of a set is a collection of sets whose union contains as a subset. Formally, if
is an indexed family of sets , then is a cover of if
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 Uα of X whose union is the whole space X. In this case we say that C covers X, or that the sets Uαcover 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 open cover if each of its members is an open set (i.e. each Uα 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 which intersects only finitely many sets in the cover. Formally, C = {Uα} is locally finite if for any x ∈ X, there exists some neighborhood N(x) of x such that the set
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