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Specialty Definition: Cover

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Specialty Definition: Cover

(From Wikipedia, the free Encyclopedia)

In telecommunication, cover is the technique of concealing or altering the characteristics of communications patterns for the purpose of denying an unauthorized receiver information that would be of value.

Note: Cover is a process of modulo two addition of a pseudorandom bit stream generated by a cryptographic device with bits from the control message.

Source: from Federal Standard 1037C and from MIL-STD-188 In philately a cover is an envelope or package, typically with stamps that have been cancelled. In pop music a cover is a new version of a previously recorded song. Virtually all pop musicians play covers, as tributes to their mentors, or on the theory that what was a hit before may be a hit again, or to gain credibility from their comparison with the original. See also: cover version. A cover band may play only music by one more prominent band or may play music from many sources. According to the Oxford English Dictionary, the original sense of the verb and noun cover was "to hide from view" as in its cognate covert. Except in the limited sense of "cover again", the word recover is unrelated and is cognate with recuperate.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Cover."

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Cover version

(From Wikipedia, the free Encyclopedia)

A cover version of a pop song is a rerecording of that song by a different artist (compare with remake).

From early in the 20th century it was common practice among phonograph record labels that if any company had a record that was a significant commercial success, other record companies would have singers or musicians "cover" the tune by recording a version for their own label in hopes of cashing in on the tune's success.

In the early days of rock and roll, many songs originally recorded by African American rock musicians were rerecorded by white artists, such as Pat Boone, in a more toned down style that lacked the hard edge of rock and roll. These cover versions were considered by some to be more palatable to parents, and white artists were more palatable to programmers at white radio stations.

Over the years, cover versions of many popular songs have been recorded, sometimes with a radically different style, and in other cases the cover version is virtually indistinguishable from the original. For example, Jose Feliciano's version of "Light My Fire" was utterly distinct from the original version by The Doors; but Carl Carlton's 1974 cover of Robert Knight's 1967 hit single song "Everlasting Love" sounds almost identical to the original. Cover versions can also be in different languages; for example, Falco's 1982 German-language hit "Der Kommissar" was covered in English by After the Fire later in the decade, although the German title was retained. The English version, which was not a direct translation of Falco's original but retained much of its spirit, reached the Top 5 on the US charts.

The subgenre of the cover version that existed from the early 1950's to the late 1970's in Louisiana was known as Swamp Pop. Contemporary and classic rock, R&B, and country songs were re-recorded with Cajun preferences in mind. Some lyrics were translated to French, and some were recorded with traditional Cajun instrumentation. Several Swamp Pop originals charted nationally, but it was mostly a regional niche market.

Punk music is known for deconstructing classic rock or pop songs by reinterpreting them in punk form. Bands like Me First & the Gimme Gimmes, The Mighty Mighty Bosstones, NOFX and Goldfinger are especially known for doing so. In recent years, several jam bands and related groups have begun covering hip hop songs, most frequently only live in concert. Perhaps the most famous such-cover recorded in a studio and released commercially is a bluegrass version of "Gin and Juice" by Snoop Doggy Dogg, as performed by the Gourds. Other artists like Phish and Keller Williams have covered "Rappers Delight" (The Sugarhill Gang), "I Like Big Butts" (Sir Mix-A-Lot) and other hip hop songs.

The Beatles have been covered more than any other band; "Yesterday" has been covered over three thousand times since its original release in 1965. Other songs which have been released many times as cover versions include the infamous "Louie Louie" by Richard Berry, "Free Bird" (Lynyrd Skynyrd), "No Woman No Cry" (Bob Marley & the Wailers) and many of the less recent works of Bob Dylan and Leonard Cohen (as of December 31, 2002, there were at least 759 published cover versions of Cohen songs [1]).

Many popular bands have a tribute album, consisting entirely of covers of their songs performed by various other bands, often quite different from the original. The soundtrack to the film I Am Sam was a particularly popular example of this; it consisted of Beatles songs redone by various modern artists.

Jam bands such as Phish and The Grateful Dead are known for playing covers in concert, generally, and not on studio albums.

Some examples of commercially successful or otherwise notable cover versions are:

Song	Original Artist, link to year in music	Cover Artist, link to year in music
"Across the Universe"	The Beatles, 1970	Fiona Apple, 1998 (from Pleasantville)
"After Midnight"	J.J. Cale, 1965	Eric Clapton, 1970 Eric Clapton, 1988 (remake)
"Ain't That a Shame"	Fats Domino, 1955	Pat Boone, 1955
"Ain't Too Proud to Beg"	The Temptations, 1966	The Rolling Stones, 1974
"Alison"	Elvis Costello, 1977	Linda Ronstadt, 1978
"All Along The Watchtower"	Bob Dylan, 1968	Jimi Hendrix, 1968
"Almost Cut My Hair"	Crosby, Stills & Nash, 1970	The Dayglo Abortions, 1998
"American Pie"	Don McLean, 1971	Madonna, 2000
"American Woman"	The Guess Who, 1970	Lenny Kravitz, 2000
"Angel in the Morning"	Merrilee Rush, 1968	Juice Newton, 1981
"Back in the USA"	Chuck Berry, 1959	Linda Ronstadt, 1978
"Because the Night"	Patti Smith, 1978 (cowritten with Bruce Springsteen)	10,000 Maniacs, 1994
"Black Magic Woman"	Fleetwood Mac, 1969	Santana, 1970
"Blinded by the Light"	Bruce Springsteen, 1973	Manfred Mann, 1976
"Blue Bayou"	Roy Orbison, 1963	Linda Ronstadt, 1977
"Blue Monday"	New Order, 1983	Orgy, 1998
"Blue Suede Shoes"	Carl Perkins, 1955	Elvis Presley, 1956
"Boogie Woogie Bugle Boy"	The Andrews Sisters, 1941	Bette Midler, 1972
"Boy From New York City"	Ad Libs, 1965	Manhattan Transfer, 1981
"Calling Occupants of Interplanetary Craft"	Klaatu, 1976	The Carpenters, 1977
"Come Together"	The Beatles, 1969	Aerosmith, 1978
"Da Doo Run Run"	The Crystals 1963	Shaun Cassidy 1977
"Dancing in the Street"	Martha & the Vandellas, 1964	Van Halen, 1982
"Daydream Believer"	The Monkees, 1968	Anne Murray, 1978 Shonen Knife, 1998
"Downtown Train"	Tom Waits, 1985	Rod Stewart, 1989
"Everlasting Love"	Robert Knight, 1967	Carl Carlton, 1974
"Everything I Own"	Bread, 1972	Ken Boothe, 1974 Boy George, 1987
"Feelin' Alright"	Traffic, 1968	Joe Cocker, 1969
"Get Ready"	The Temptations, 1966	Rare Earth, 1970
"Gin and Juice"	Snoop Doggy Dogg, 1993	The Gourds, 1998
"Gloria"	Them (with Van Morrison), 1965	Shadows of Knight, 1966
"A Hard Day's Night"	The Beatles, 1964	Goldie Hawn, 1998
"A Hazy Shade Of Winter"	Simon and Garfunkel, 1966	The Bangles, 1987
"Heat Wave"	Martha & the Vandellas, 1963	Linda Ronstadt, 1975
"Helter Skelter"	The Beatles, 1968	U2, 1989
"Hey Baby"	Bruce Channel, 1962	DJ Ötzi, 2000
"Hurt"	Nine Inch Nails, 1994	Johnny Cash, 2002
"I Don't Like Mondays"	The Boomtown Rats, 1982	Tori Amos, 2001
"I Fought the Law"	Bobby Fuller Four, 1966	The Clash, 1979
"I Got Rhythm"	George Gershwin Broadway musical Girl Crazy, 1930	The Happenings, 1967
"I Got You Babe"	Sonny & Cher, 1967	Cher with Beavis and Butthead, 1993
"I Hear You Knocking"	Smiley Lewis, 1961	Dave Edmunds, 1971
"I Heard It Through The Grapevine"	Gladys Knight & the Pips, 1967	Marvin Gaye, 1968 Creedence Clearwater Revival, 1970
"I Just Don't Know What to Do With Myself"	Dusty Springfield, 1964	Elvis Costello, 1978
"I Put a Spell on You"	Screaming Jay Hawkins, 1957	Creedence Clearwater Revival, 1968 Marilyn Manson, 1995
"I Shot the Sheriff"	Bob Marley & the Wailers, 1973	Eric Clapton, 1974
"If"	David Gates, 1971	Telly Savalas, 1975
"I'm a Believer"	Neil Diamond, 1966	The Monkees, 1966 Smashmouth, 2001
"I'm a Man"	Spencer Davis Group, 1967	Chicago, 1970
"In the Midnight Hour"	Wilson Pickett, 1965	The Rascals, 1967
"It Ain't Me Babe"	Bob Dylan, 1964	The Turtles, 1965
"Jealous Guy"	John Lennon, 1971	Roxy Music, 1981
"Killing Me Softly With His Song"	Roberta Flack, 1973	The Fugees, 1996
"Knocking on Heaven's Door"	Bob Dylan, 1973	Guns N' Roses, 1991
"Last Kiss"	J. Frank Wilson, 1964	Pearl Jam, 1981
"The Letter"	The Boxtops, 1967	Joe Cocker, 1970
"Light My Fire"	The Doors, 1967	Jose Feliciano, 1968
"The Lion Sleeps Tonight"	The Tokens, 1961	Robert John, 1971 Tight Fit, 1982
"Live and Let Die"	Paul McCartney, 1976	Guns N' Roses, 1991
"The Locomotion"	Little Eva, 1962	Grand Funk, 1974
"Love is All Around"	The Troggs, 1967	Wet Wet Wet, 1994
"Love is Strange"	Ian and Sylvia, 1958	Peaches and Herb, 1967
"Love Potion Number 9"	The Clovers, 1959	The Searchers, 1964
"Lover's Cross"	Jim Croce, 1972	Melanie, 1974
"Lucy in the Sky with Diamonds"	The Beatles, 1967	William Shatner, 1968 Elton John, 1974
"MacArthur Park"	Richard Harris, 1968	Donna Summer, 1978
"The Man Who Sold The World"	David Bowie, 1970	Nirvana, 1994
"Mr. Tambourine Man"	Bob Dylan, 1965	The Byrds, 1965
"My Way"	Frank Sinatra, 1969	Sid Vicious, 1978
"No Woman, No Cry"	Bob Marley & the Wailers, 1974	Gilberto Gil, 1980
"Ob-La-Di, Ob-La-Da"	The Beatles, 1968	The Marmalade, 1968
"Only You"	Yazoo, 1982	Flying Pickets, 1983
"Pinball Wizard"	The Who, 1969	The New Seekers, 1973 Elton John, 1976
"Proud Mary"	Creedence Clearwater Revival, 1969	Ike and Tina Turner, 1971
"Rock and Roll Music"	Chuck Berry, 1957	The Beatles, 1964 The Beach Boys, 1975
"Roll Over Beethoven"	Chuck Berry, 1956	The Beatles, 1963 Electric Light Orchestra, 1973
"Ruby Tuesday"	The Rolling Stones, 1967	Melanie, 1972
"Shameless"	Billy Joel, 1989	Garth Brooks, 1991
"She Came in through the Bathroom Window"	The Beatles, 1969	Joe Cocker, 1969
"Somethin' Stupid"	Frank & Nancy Sinatra, 1967	Robbie Williams & Nicole Kidman, 2001
"Something in the Air"	Thunderclap Newman, 1969	Fish, 1991
"Spanish Harlem"	Ben E. King, 1961	Aretha Franklin, 1971
"Stay"	Maurice Williams & the Zodiacs, 1960	Jackson Browne, 1977
"Stop Your Sobbing"	The Kinks, 1964	The Pretenders, 1979
"Summertime"	George Gershwin, from the opera Porgy and Bess, 1934	The Shake Spears, 1966 Janis Joplin, 1967 [1]
"Summertime Blues"	Eddie Cochran, 1958	The Who, 1967 Blue Cheer, 1968
"Suzie Q"	Dale Hawkins, 1957	Creedence Clearwater Revival, 1968
"Suspicion"	Elvis Presley, 1962	Terry Stafford, 1964
"Sweet Jane"	Velvet Underground, 1970	Cowboy Junkies, 1988
"Sympathy for the Devil"	Rolling Stones, 1968	Jane's Addiction, 1987
"Take Me To The River"	Al Green, 1974	Talking Heads, 1978
"That'll Be The Day"	Buddy Holly, 1957	Linda Ronstadt, 1976
"There She Goes"	The La's, 1989	Sixpence None The Richer, 1999
"This Magic Moment"	The Drifters, 1960	Jay and the Americans, 1969
"Time"	Tom Waits, 1967	Rod Stewart, 1970
"Top of the World"	The Carpenters, 1972	Shonen Knife, 1994
"The Tracks of My Tears"	Smokey Robinson & the Miracles, 1965	Johnny Rivers, 1967 Linda Ronstadt, 1976
"Twist and Shout"	The Isley Brothers, 1962	The Beatles, 1963 (rereleased 1986)
"Venus"	Shocking Blue, 1969	Bananarama, 1986
"Walk on By"	Dionne Warwick, 1964	Isaac Hayes, 1969
"Walk This Way"	Aerosmith, 1975	Run DMC, 1986
"We Can Work It Out"	The Beatles, 1966	Stevie Wonder, 1971
"Wild Night"	Van Morrison, 1971	John Cougar Mellencamp & Me'Shell NdegeOcello, 1994
"Wild World"	Cat Stevens, 1970	Jimmy Cliff, 1970
"With a Little Help from My Friends"	The Beatles, 1967	Joe Cocker, 1969
"Without You"	Badfinger, 1970	Harry Nilsson, 1971
"Woodstock"	Joni Mitchell, 1970	Crosby, Stills, and Nash, 1970
"You Keep Me Hanging On"	The Supremes, 1966	Vanilla Fudge, 1968
"You Really Got Me"	The Kinks, 1965	Brian Eno et al 1974, Van Halen, 1978
"You've Got a Friend"	Carole King, 1971	James Taylor, 1971 [1]

Artists that have released albums consisting entirely of cover songs include:

• The Afghan Whigs - Uptown Avondale (EP, 1992)
• Tori Amos - Strange Little Girls (2001)
• Big Country - Undercover (2001)
• Pat Boone - In a Metal Mood: No More Mr. Nice Guy (1997)
• David Bowie - Pin-Ups (1973)
• Elvis Costello - Almost Blue (1981)
• Elvis Costello - Kojak Variety (1995)
• Duran Duran - Thank You (1995)
• Erasure - Other People's Songs (2003)
• Guana Batz - Undercover (1999)
• Joe Jackson - Jumpin' Jive (1981)
• Cyndi Lauper - At Last (2003)
• John Mellencamp - Trouble No More (2003)
• Metallica - Garage Inc
• Rage Against the Machine - Renegades (2000)
• Trixter - UnderCovers (1994)
• Dwight Yoakam - Under the Covers (1997)
• Yo La Tengo - Fakebook

An album consisting entirely of covers of the same artist is called a tribute album. Notable tribute albums include:

• Almost You: The Songs of Elvis Costello
• Stairways to Heaven (22 covers of Led Zeppelin's "Stairway to Heaven"}
• Tales of Wonder - Nnenna Freelon (covers of Stevie Wonder songs)
• Labour of Love: The Music of Nick Lowe

Artists such as Weird Al Yankovic produce parodies of popular songs, lampooning the originals by changing aspects of it, from lyrics to instrumentation. See Dr. Demento

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Cover version."

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Covering map

(From Wikipedia, the free Encyclopedia)

In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property:

to every x∈X there exists an open neighborhood U such that p ^-1(U) is a union of mutually disjoint open sets S_i (where i ranges over some index set I) such that p restricted to S_i yields a homeomorphism from S_i to U for every i∈I.

A covering map is also simply called a cover; we say C is a covering space of X or C covers X. For each x∈X, the set p ^-1(x'\') is called the fiber over x; the sets Si are called the sheets over U. One generally pictures C as "hovering above" X, with p mapping "downwards", the sheets over U being horizontally stacked above each other and above U, and the fiber over x consisting of those points of C that lie "vertically above" x''.

Examples

Consider the unit circle S¹ in R². Then the map p : R → S¹ with p(t) = (cos(t),sin(t)) is a cover.

Consider the complex plane with the origin removed, denoted by C^×, and pick a non-zero integer n. Then p : C^× → C^× given by p(z) = zⁿ is a cover. Here every fiber has n elements.

If G is group (considered as a discrete topological group), then every principal G-bundle is a covering map. Here every fiber can be identified with G.

Elementary properties

Every cover p : C → X is a local homeomorphism (i.e. to every c∈C there exists an open set A in C containing c and an open set B in X such that the restriction of p to A yields a homeomorphism between A and B). This implies that C and X share all local properties.

For every x∈X, the fiber over x is a discrete subset of C. On every connected component of X, the cardinality of the fibers is the same (possibly infinite). If every fiber has 2 elements, we speak of a double cover.

The lifting property: if p : C → X is a cover and γ is a path in X (i.e. a continuous map from the unit interval [0,1] into X) and c∈C is a point "lying over" γ(0) (i.e. p(c) = γ(0)), then there exists a unique path ρ in C lying over γ (i.e. p o ρ = γ) and with ρ(0) = c.

If x and y are two points in X connected by a path, then that path furnishes a bijection between the fiber over x and the fiber over y via the lifting property.

Universal covers

A cover q : D → X is a universal cover iff D is simply connected. The name comes from the following important property: if p : C → X is any cover of X with C connected, then there exists a covering map f : D → C such that p o f = q. This can be phrased as "The universal cover of X covers all connected covers of X."

The map f is unique in the following sense: if we fix x∈X and d∈D with q(d) = x and c∈C with p(c) = x, then there exists a unique covering map f : D → C such that p o f = q and f(d) = c.

If X has a universal cover, then that universal cover is essentially unique: if q₁ : D₁ → X and q₂ : D₂ → X are two universal covers of X, then there exists a homeomorphism f : D₁ → D₂ such that q₂ o f = q₁.

The space X has a universal cover if and only if it is path-connected, locally path-connected and semi-locally simply connected. The universal cover of X can be constructed as a certain space of paths in X.

The example R → S¹ given above is a universal cover. The map S³ → SO(3) from unit quaternions to rotations of 3D space described in quaternions and spatial rotation is also a universal cover.

If the space X carries some additional structure, then its universal cover normally inherits that structure:

• if X is a manifold, then so is its universal cover C
• if X is a Riemann surface, then so is its universal cover C, and p is a holomorphic map
• if X is a Lie group (as in the two examples above), then so is its universal cover C, and p is a homomorphism of Lie groups.

Deck transformation group, regular covers

A deck transformation or automorphism of a cover p : C → X is a homeomorphism f : C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut(p).

Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber.

Now suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. The action of Aut(p) on each fiber is free. If this action is transitive on some fiber, then it is transitive on all fibers, and we call the cover regular. Every such regular cover is a principal G-bundle, where G = Aut(p) is considered as a discrete topological group.

Every universal cover p : D → X is regular, with deck transformation group being isomorphic to the opposite of the fundamental group π(X).

The example p : C^× → C^× with p(z) = zⁿ from above is a regular cover. The deck transformations are multiplications with n-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group C_n.

Monodromy action

Again suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. If x∈X and c belongs to the fiber over x (i.e. p(c) = x), and γ:[0,1]→X is a path with γ(0)=γ(1)=x, then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x. It turns out that this end point only depends on the class of γ in the fundamental group π(X,x), and in this fashion we obtain a right group action of π(X,x) on the fiber over x. This is known as the monodromy action.

So there are two actions on the fiber over x: Aut(p) acts on the left and π(X,x) acts on the right. These two actions are compatible in the following sense:

f.(c.γ) = (f.c).γ

for all f∈Aut(p), c∈p ^-1(x) and γ∈π(X,x).

If p is a universal cover, then the monodromy action is regular; if we identify Aut(p) with the opposite group of π(X,x), then the monodromy action coincides with the action of Aut(p) on the fiber over x.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Covering map."

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Topology glossary

(From Wikipedia, the free Encyclopedia)

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no clear distinction between different areas of topology, this glossary focuses primarily on general topology and definitions that are fundamental to a broad range of areas. See the article on topological spaces for basic definitions and examples, and see the article on topology for a brief history and description of the subject area.

The following articles may also be useful. These either contain specialised vocabulary within general topology or provide more detailed expositions of the definitions given below. The list of general topology topics will also be very helpful.

• Compact space
• Connected space
• Continuity (topology)
• Metric space
• Separated sets
• Separation axiom
• Uniform space

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.

• Accessible. See T₁.

• Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense.

• Base. A set of open sets is a base (or basis) for a topology if every open set in the topology is a union of sets in the base. The topology generated by a base is the smallest topology containing the base elements; this topology consists of all unions of elements of the base.

• Basis. See Base.

• Borel algebra. The Borel algebra on a space X is the smallest σ-algebra containing all the open sets.

• Borel set. A Borel set is an element of a Borel algebra.

• Boundary. The boundary of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement.

• Cauchy sequence. A sequence {x_i} in a metric space M with metric d is called a Cauchy sequence (or Cauchy for short) if for every positive real number r, there is an integer N such that for all integers m and n greater than N, the distance d(x_m, x_n) is less than r.

• Clopen. A set is clopen if it is both open and closed.

• Closed set. A set is closed if its complement is a member of the topology.

• Closed function. A function from one space to another is closed if the image of every closed set is closed.

• Closure. The closure of a set is the intersection of all closed sets which contain it. It is the smallest closed set containing the original set.

• Compact. A space is compact if every open cover has a finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.

• Complete. A metric space is complete if every Cauchy sequence converges.

• Completely metrizable/completely metrisable. See Topologically complete.

• Completely normal. A space is completely normal if any two separated sets have disjoint neighbourhoods.

• Completely normal Hausdorff. A completely normal Hausdorff space (or T₅ space) is a completely normal T₁ space. (A completely normal space is Hausdorff if and only if it is T₁, so the terminology is consistent.) Completely normal Hausdorff spaces are always normal Hausdorff.

• Completely regular. A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are functionally separated.

• Completely regular Hausdorff. See Tychonoff.

• Completely T₃. See Tychonoff.

• Component. See connected component.

• Connected. A space X is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.

• Connected component. A connected component of a space is a maximal connected subspace. The connected components of a space form a partition of that space.

• Continuous. A function from one space to another is continuous if the preimage of every open set is open.

• Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.

• Countably compact. A space is countably compact if every countable open cover has a finite subcover.

• Cover. A collection {U_i} of sets is a cover (or covering), if their union is the whole space. An open cover is a cover consisting of open sets.

• Covering. See Cover.

• Dense. A dense set is a set that meets every nonempty open set in the space. Equivalently, a set is dense if its closure is the whole space.

• Discrete topology. See Discrete space.

• Discrete space. A space X is discrete if every set is open. We say that X carries the discrete topology.

• Entourage. See Uniform space.

• F_σ set. An F_σ set is a countable union of closed sets.

• First category. See Meagre.

• First-countable. A space is first-countable if every point has a countable local base.

• Functionally separated. Two sets A and B in a space X are functionally separated if there is a continuous function from X into the interval [0,1] with the property that A is mapped to 0 and B is mapped to 1.

• G_δ set. A G_δ set is a countable intersection of open sets.

• Hausdorff. A space is Hausdorff (or T₂) if every two distinct points have disjoint neighbourhoods. Hausdorff spaces are always T₁.

• Hereditary. A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it. For example, second-countability is a hereditary property.

• Homeomorphism. A homeomorphism from a space X to a space Y is a bijective map f : X → Y such that f and f ^-1 are continuous. The spaces X and Y are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical.

• Homogeneous. A space X is homogeneous if for every x and y in X there is a homeomorphism f : X -> X such that f(x) = y. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous.

• Homotopic maps. Two continuous maps f, g : X -> Y are homotopic if there is a continuous map H: X× [0,1] → Y, such that H(x,0) = f(x) and H(x,1) = g(x) for all x in X. Here, the space X × [0,1] is given the usual product topology. The function H is called a homotopy between f and g.

• Indiscrete space. See Trivial topology.

• Indiscrete topology. See Trivial topology.

• Interior. The interior of a set is the union of all open sets contained in it. It is the largest open set contained in the original set.

• Isolated point. A point x is an isolated point if the singleton {x} is open.

• Kolmogorov. See T₀.

• Kuratowski closure axioms. The Kuratowski closure axioms are a set of axioms satisied by the closure operator:

Isotonicity: Every set is contained in its closure.

Idempotence: The closure of the closure of a set is equal to the closure of that set.

Preservation of binary unions: The closure of the union of two sets is the union of their closures.

Preservation of nullary unions: The closure of the empty set is empty.

• Limit point. A point x in X is a limit point of a subset S if every open set containing x also contains a point of S other than x itself. This is equivalent to requiring that every neighbourhood of x contains a point of S other than x itself.

• Lindelöf. A space is Lindelöf if every open cover has a countable subcover.

• Local base. A set B of neighbourhoods of a point x of a topological space X is a local base (or local basis, neighbourhood base, neighbourhood basis) at x if every neighbourhood of x contains some member of B.

• Local basis. See Local base.

• Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Locally compact Hausdorff spaces are always Tychonoff.

• Locally connected. A space is locally connected if every point has a local base consisting of connected sets.

• Locally finite. A collection of subsets of a space is locally finite if every point has a neighbourhood which meets only finitely many of the subsets.

• Locally metrizable/Locally metrisable. A space is locally metrizable if every point has a metrizable neighbourhood.

• Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.

• Meagre. If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable union of nowhere dense sets. If A is not meagre in X, A is sometimes said to be of second category in X.

• Metric. See Metric space.

• Metric space. A metric space is a set M equipped with a function d : M × M → R satisfying the following conditions for all x, y, and z in M:

d(x, y) ≥ 0

d(x, x) = 0

if   d(x, y) = 0   then   x = y     (identity of indiscernibles)

d(x, y) = d(y, x)     (symmetry)

d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality)

The function d is called a metric on M.

• Metrizable/Metrisable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable.

• Neighbourhood/Neighborhood. A neighbourhood of a set S is a set containing an open set which in turn contains the set S. (Note that the neighbourhood itself need not be open.) A neighbourhood of a point p is a neighbourhood of the singleton set {p}.

• Neighbourhood base. See Local base.

• Neighbourhood basis. See Local base.

• Net. A net in a space X is a map from a directed set A to X. A net from A to X is usually denoted (x_α), where α is in an index variable ranging over A. Every sequence is a net, taking A to be the directed set of natural numbers with the usual ordering.

• Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity.

• Normal Hausdorff. A normal Hausdorff space (or T₄ space) is a normal T₁ space. (A normal space is Hausdorff if and only if it is T₁, so the terminology is consistent.) Normal Hausdorff spaces are always Tychonoff.

• Nowhere dense. A nowhere dense set is a set whose closure has empty interior.

• Open cover. See Cover.

• Open set. A set is open if it is a member of the topology.

• Open function. A function from one space to another is open if the image of every open set is open.

• Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.

• Partition of unity. A partition of unity of a space X is a set of continuous functions from X to [0,1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.

• Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected.

• Point. This term is often used to refer to elements of the topological space.

• Polish. A space is called Polish if it is metrizable with a separable and complete metric.

• Product topology. If {X_i} is a collection of spaces and X is the (set-theoretic) product of {X_i}, then the product topology on X is the weakest topology for which all the projection maps are continuous.

• Punctured neighbourhood/Punctured neighborhood. A punctured neighbourhood of a point p is a neighbourhood of p, minus {p}. For instance, the interval (-1,1) = {x : -1 < x < 1} is a neighbourhood of 0 in the real line, so the set (-1,0) ∪ (0,1) = (-1,1) - {0} is a punctured neighbourhood of 0.

• Quotient space. If X and Y are spaces and f : X → Y is any function, then the quotient space on Y induced by f is the weakest topology for which f is continuous. The most common example of this is to consider an equivalence relation on X, with Y the set of equivalence classes and f the natural projection map.

• Refinement. A cover K is a refinement of a cover L if every member of K is a subset of some member of L.

• Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.

• Regular Hausdorff. A space is regular Hausdorff (or T₃) if it is a regular T₀ space. (A regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.)

• Residual. If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X.

• Second category. See Meagre.

• Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.

• Separable. A space is separable if it has a countable dense subset.

• Separated. Two sets A and B are separated if each is disjoint from the other's closure.

• Sierpinski space. Let S = {0,1}. Then T = is a topology on S, and the resulting space is called Sierpinski space. The Sierpinski space is the simplest example of a space that does not satisfy the T₁ axiom.

• Simply connected. A space X is simply connected if it is path-connected and every continuous map f: S¹ → X is homotopic to a constant map.

• Subbase. A set of open sets is a subbase (or subbasis) for a topology if every open set in the topology is a union of finite intersections of sets in the subbase. The topology generated by a subbase is the smallest topology containing the subbase elements; this topology consists of all finite intersections of unions of elements of the subbase.

• Subbasis. See Subbase.

• Subcover. A cover K is a subcover (or subcovering) of a cover L if every member of K is a member of L.

• Subcovering. See Subcover.

• Subspace. If X is a space and A is a subset of X, then the subspace topology on A induced by X consists of all intersections of open sets in X with A.

• T₀. A space is T₀ (or Kolmogorov) if for every pair of distinct points x and y in the space, either there is an open set containing x but not y, or there is an open set containing y but not x.

• T₁. A space is T₁ (or accessible) if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T₀; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T₁ if all its singletons are closed. T₁ spaces are always T₀.

• T₂. See Hausdorff.

• T₃. See Regular Hausdorff.

• T_3½. See Tychonoff.

• T₄. See Normal Hausdorff.

• T₅. See Completely normal Hausdorff.

• Topological space. A topological space is a set X equipped with a collection T of subsets of X satisfying the following conditions:

The empty set and X are in T.

The union of any collection of sets in T is also in T.

The intersection of any pair of sets in T is also in T.

The collection T is called a topology on X.

• Topologically complete. A space is topologically complete if it is homeomorphic to a complete metric space.

• Topology. See Topological space.

• Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.

• Trivial topology. The trivial topology on a set X consists of precisely the empty set and the entire space X.

• Tychonoff. A Tychonoff space (or completely regular Hausdorff space, completely T₃ space, T_3½ space) is a completely regular T₀ space. (A completely regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.

• Uniform space. A uniform space is a set U equipped with a nonempty system Φ of subsets of the Cartesian product X × ''X'\' satisfying the following:

if U is in Φ, then U contains { (x, x) : x in X }.

if U is in Φ, then { (y, x) : (x, y) in U } is also in Φ

if U is in Φ and V is a subset of X × X which contains U, then V is in Φ

if U and V are in Φ, then U ∩ V is in Φ

if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in U.

The elements of Φ are called entourages, and Φ itself is called a uniform structure on U.

• Uniform structure. See Uniform space.

• Weak topology. The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest topology on the set which makes all the functions continuous.

• Weakly hereditary. A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary.

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Topology glossary."

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Photo Album: Cover

Thumbnail	Description & Credit	Thumbnail	Description & Credit
	Shown is an outdoor summer scene in a treed area with a lake visible. Also seen are two runners in shorts. This is the cover photograph to the "Decade of Discovery" section entitled, "Lifestyle, Environment and Cancer". Credit: Linda Bartlett (photographer).		From an overhead angle, bread, chinese cabbage, strawberry, grapes and brussels sprouts are shown on a white patterned tabletop. On the purple cloth in the top left, white letters read: "Eat for Good Health". Shot on 4x5 format. This was used in the 1989 calendar "Eat for Good Health" Cover 1989. See artwork: PV-19. Credit: Bill Branson (photographer).
	Photograph of cover of 1992 Institute of Medicine Report, Emerging Infections: Microbial Threats to Health in the United States. Credit: CDC.		Host cell plasma membrane derived from the previous host cell may cover the rickettsia. Transmission electron micrograph. Credit: CDC.
	The Sounds of Earth Record Cover. Credit: NASA.		Who's Who - Cover sheet to 1931 C&GS publication Shows all former C&GS Superintendents/Directors Book includes autobiographical sketches of about 200 C&GS employees. Credit: Coast & Geodetic Survey Historical Image Collection.
	At the head of Walker Cover. Credit: Coast & Geodetic Survey Historical Image Collection.		Pinfish seeking cover in mangrove roots. Credit: America's Coastlines.
	The cover to "Puget Sound and Western Washington Cities-Towns Scenery", by Robert A. Reid, Robert A. Reid Publisher, Seattle, 1912. Credit: America's Coastlines.		Strolling down main street of the Oliktok Point Camp Snow plows cover entrances to buildings similar to more urban areas. Credit: Paths Less Taken - NOAA at the Ends of the Earth.
Source: pictures compiled by the editor from various references; see picture credits.

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Digital Photo Gallery: Cover
 

"Cover baby" by Jp Vooys Commentary: "The perfect magazine portrait of an infant."	"Tosca libretto cover" by Pedro Valdeolmillos Commentary: "Cover detail of Tosca opera libretto."
Source: photographs selected by the editor, with permission from the photographers.

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