In topology , the longline or Alexandroff line is a topological space somewhat similar to the real line ... in the sense that it defines the order on the longline and shows that the topology is the order ... consists of a countable number of line segments nowiki nowiki 0, 1 laid end to end, whereas the longline is constructed from an uncountable number of such segments. Definition The closed long ray ... . The longline is obtained by putting together a long ray in each direction. More rigorously, it can be defined as the order topology on the disjoint union of the reversed open long ray &ldquo reversed&rdquo ... t of the other, and define the longline to be the topological space obtained by gluing the two ... from the topological point of view. Intuitively, the closed long ray is like a real closed half line .... The open long ray is like the real line or equivalently an open half line except that it is much ... line is longer than the real lines in both directions we say that it is long in both directions. However ... of the line, and it doesn t matter what happens at the other end whether long, short, or closed . A related ... longline by adding two elements to the longline, one at each end. Properties The closed long ray L ... properties in their manifolds do not call the longline a manifold Citation needed date June 2011 . The longline or ray can be equipped with the structure of a non separable differentiable manifold ... C sup k 1 sup or C sup &infin sup structures on the longline or ray inducing any given C sup k ... interval real line , the half open interval, the closed long ray, the open long ray, or the longline Citation needed date June 2011 . The longline or ray can even be equipped with the structure of a real ... year 1960 pages 104&ndash 106 doi 10.1007 BF01236917 ref The longline or ray cannot be equipped with a Riemannian ... compactification , because any continuous function from the closed or open long ray to the real line ... because the longline is too long to be covered by a path, which is a continuous image of an interval ... more details
Longline or longline may refer to LongLine , an album by Peter Wolf Longlinetopology , or Alexandroff line, a topological space Longline telecommunications , a transmission line in a long distance communications network Longline fishing , a commercial fishing technique AT&T Long Lines , a telecommunications network Aerial crane or longline , a transport method Disambiguation ... more details
About the Peter Wolf album other uses Longline disambiguation Infobox Album See Wikipedia WikiProject Albums Name LongLine Cover Peter Wolf Long Line.jpg Type Album Artist Peter Wolf Released May 14, 1996 Recorded Longview Studios, North Brookfield, MA vocals recorded at Profit Sound, Stoughton, MA Genre Rock music Rock Length 47 13 Label Reprise Records Reprise Producer Peter Wolf , br Johnny A. , br Stu Kimball Last album Up to No Good br 1990 This album LongLine br 1996 Next album Fool s Parade br 1998 Album ratings rev1 Allmusic AMG rev1Score Rating 3.5 5 ref Allmusic class album id r235397 review ref rev2 Rolling Stone rev2Score Rating 4 5 ref http www.rollingstone.com artists peterwolf albums album 222768 review 5944622 longline ref LongLine is the fourth solo album by Peter Wolf , released in 1996 see 1996 in music . Track listing LongLine Angelo Petraglia, Stu Kimball, Wolf 3 33 Romeo Is Dead Will Jennings , Wolf 3 24 Rosie Taylor Rhodes, Wolf 4 26 Forty to One Aimee Mann , Wolf 3 00 Goodbye Is All I ll Send Her Kimball, Wolf 3 54 Wastin Time Wolf 4 16 Sky High Jennings, Wolf 3 45 Two Loves Wolf 3 30 Break This Chain Robert White Johnson, Rhodes, Wolf 4 46 Seventh Heaven Johnson, Rhodes, Wolf 3 47 Starvin to Death Mann, Wolf 3 27 Riverside Drive Wolf 5 25 Personnel Peter Wolf harmonica , Singing vocals Beth A. background vocals Johnny A . guitar s, baritone guitar Tim Archibald bass guitar bass Patty Barkas vocals, background vocals Tom Belliveau pedal steel Brad Delp Bradley Delp background vocals Doug Dube organ music organ Charlie Farren background vocals Ken Field flute , saxophone Stuart Kimball organ, guitar Matt Leavenworth fiddle Brian Maes keyboard instrument keyboard , background vocals Holly Palmer background vocals Tom Soares background vocals David Stefanelli percussion instrument percussion , Drum kit drums Bird Taylor background vocals Buck Taylor background vocals Bob Tudor keyboard Production Producers Peter Wolf, Johnny A., Stu Kimball ... more details
Unreferenced date December 2009 In telephone systems nomenclature, a longline is a transmission line in a long distance communications network such as carrier system s, microwave radio relay links, geosynchronous satellite links, underground cables, aerial cables and open wire, and Submarine communications cable s. In the Communications in the United States United States , some of this technology was spun off into the corporate entity known as AT&T Long Distance with the breakup of AT&T in 1984. Prior to then, the AT&T Long Lines division of the Bell System , which was what AT&T called itself when it was responsible for providing both long distance and local telephone service before the Bell System divestiture 1984 breakup , provided maintenance and installation of longline facilities for the Bell System s long distance service. See also Long haul communications Long distance calling Telecommunications DEFAULTSORT LongLine Telecommunications Category Telephony Category Communication circuits ... more details
Longline rider was a person in the Arkansas prison system of the twentieth century who supervised inmates, particularly in the area of field production. He was second in command to the inmate yardman. ref Terror In The Prisons Homosexual Rape and Why Society Condones It , Carl Weiss & David James Friar, Bobbs Merrill , 1974, p.231. ref Bobby Darin s song LongLine Rider ref Bobby Darin The Incredible Story of an Amazing Life , Al DiOrio, Running Press, May 11 , 2004 , p.175. ref highlighted the terror of the Cummins Prison Farm Cummins Unit , where the skeletons of three prisoners were discovered in January 1968. ref name joe Accomplices To The Crime The Arkansas Prison Scanadal , Tom Murton & Joe Hyams, Grove Press , p.3. ref Longline riders were armed guards who rode on horseback. They were overseers of inmates who were swinging hoes. In instances when inmates escaped, longline riders released bloodhound s to track them. ref The Other Side of Mercy A Killer s Journey Across the American Divide , Ken Armstrong & Jonathan Martin, Dog Ear Publishing, September 17, 2010, p.19. ref Longline riders at the Cummins Prison farm were made up of inmates, called trusties, who were often serving long prison sentences for armed robbery or murder. ref name joe References references Category History of Arkansas Longline rider Category People murdered in Arkansas Longline rider ... more details
Infobox Single Name LongLine of Losers Cover Cover size Border Caption Artist Montgomery Gentry Album Back When I Knew It All Released June 22, 2009 Format CD Single , music download Genre Country music Country Length 3 48 Label Columbia Records Columbia Nashville Writer Kevin Fowler br Kim Tribble Producer Blake Chancey Audio sample? Certification Last single One in Every Crowd br 2009 This single LongLine of Losers br 2009 Next single Oughta Be More Songs About That br 2009 Misc LongLine of Losers is a song recorded by United States American country music duo Montgomery Gentry . It is the fourth single released from their sixth studio album, Back When I Knew It All . Kevin Fowler who co wrote this song also recorded this song on his 2007 album Bring It On Kevin Fowler album Bring It On , also released his version as a single, but it failed to chart. Montgomery Gentry s version debuted at 48 in June 2009, was a Top 30 hit for the duo with a peak of 23 in October 2009. Reception Dan Milliken of Country Universe gave the song a B rating, and said this in his review of the song that it was their most country sounding single in some time. Milliken also said that it showed influences from Alabama band Alabama and Hank Williams, Jr. , and that its songwriting breathe d new life into a common theme of showing pride in a broken family. ref http www.countryuniverse.net 2009 06 10 montgomery gentry longline of losers ref Bobby Peacock of Roughstock also gave a positive review, calling it a good example of Montgomery Gentry s musical evolution and saying that it continued in the sense of maturity present in such singles as Back When I Knew It All song Back When I Knew It All and Roll with Me . ref cite web url http www.roughstock.com blog montgomery gentry longline of losers title Montgomery Gentry LongLine of Losers author Bobby Peacock date 2009 07 23 work Roughstock accessdate 2009 10 24 ref Chart performance LongLine of Losers debuted at 48 on the U.S. Billboard Hot ... more details
Infobox album See Wikipedia WikiProject Albums Name LongLine of Heartaches Type studio Artist Connie Smith Cover Connie Smith LongLine of Heartaches.jpg Released Start date 2011 08 23 Recorded Nashville, Tennessee Genre Country music Country Length Duration m 37 s 08 Label Sugar Hill Records Sugar Hill Producer Marty Stuart Last album Love Never Fails 2003 album Love Never Fails br 2003 This album LongLine of Heartaches br 2011 Next album LongLine of Heartaches is the thirty fourth studio album by American Country music country artist Connie Smith . It was released August 23, 2011 through Sugar Hill Records . The record is produced by Smith s husband Marty Stuart . It is her first album of new solo material since 1998, and her second only since 1978. ref cite web title Connie Smith Prepares Traditional Country Album url http www.cmt.com news news in brief 1664139 connie smith prepares traditional country album.jhtml publisher Country Music Television accessdate 23 July 2011 ref Background LongLine of Heartaches was recorded at RCA Victor Records s Studio B, where Smith cut many of her major hit singles for RCA in the 1960s and 70s. The release includes five new songs Smith co wrote with husband and producer Marty Stuart , with the remainder of the songs written by other acknowledged country songwriters, including Harlan Howard , Kostas , and Johnny Russell . ref cite web last Betts first Stephen L. title Connie Smith to Unveil a LongLine of Heartaches url http www.theboot.com 2011 05 20 connie smith longline of heartaches album publisher The Boot accessdate 23 July 2011 ref The album s official track listing was released in July through Sugar Hill Records . ref cite web title LongLine of Heartaches Connie Smith url http www.sugarhillrecords.com artists.html?controller albums&task view&id 411 publisher Sugar Hill Records accessdate 23 July 2011 ref Track listing tracklist writing credits yes title1 LongLine of Heartaches writer1 Connie Smith, Marty Stuart length1 ... more details
Infobox Film name The Long Gray Line image The Long Gray Line 1955 poster.jpg image size 300 caption 1955 Theatrical Poster director John Ford producer Robert Arthur film producer Robert Arthur writer Nardi Reeder Campion br Edward Hope br Martin Maher Marty Maher narrator starring Tyrone Power br Maureen O Hara music W. Franke Harling cinematography Charles Lawton Jr. br Charles Lang editing William A. Lyon distributor Columbia Pictures released January 22, 1955 runtime 138 min. country United States language English language English budget 1,748,000 estimated File LongGreyLineFilm.jpg 150px thumb Home video DVD cover The Long Gray Line is a 1955 American drama film directed by John Ford ref Variety Film Reviews Variety film review February 9, 1955, page 10. ref ref Harrison s Reports and Film Reviews Harrison s Reports film review February 12, 1955, page 26 ref based on the life of Martin Maher Marty Maher . Tyrone Power stars as the scrappy Irish immigrant whose 50 year career at United States Military Academy West Point took him from Dishwashing dishwasher to non commissioned officer and athletic instructor. Maureen O Hara , one of John Ford Ford s favorite leading ladies, plays Maher s wife and fellow Immigration immigrant , Mary O Donnell. The film costars Ward Bond as Herman Koehler , the Master of the Sword athletic director and Army s head football coach 1897 1900 , who first befriends Maher. Milburn Stone appears as John J. Pershing who in 1898 swears Maher into the Army. Harry Carey, Jr. makes a brief appearance as the young cadet Dwight D. Eisenhower . Philip Carey plays fictional Army football player and future general Chuck Dotson. The story follows Maher s arrival at West Point and his progress from servant to beloved leader and teacher. The film also covers ... her death. The phrase The Long Gray Line is used to describe, as a continuum, all graduates and cadets ... 0048312 Amg movie 29925 The Long Gray Line John Ford DEFAULTSORT Long Gray Line Category 1955 films ... more details
Infobox Single See Wikipedia WikiProject Songs Name A LongLine of Love Artist Michael Martin Murphey Album Americana Michael Martin Murphey album Americana A side B side Worlds Apart Released April 1987 Format Recorded ca. October 1986 Genre Country music Country Length 3 27 Label Warner Bros. Writer Paul Overstreet and Thom Schuyler Producer Jim Ed Norman Certification Last single A Face in the Crowd Michael Martin Murphey and Holly Dunn song A Face in the Crowd BR w Holly Dunn BR 1987 This single A LongLine of Love BR 1987 Next single I m Gonna Miss You, Girl BR 1987 Misc A LongLine of Love is the title of a song written by Paul Overstreet and Thom Schuyler . It was recorded by American country music artist, Michael Martin Murphey . It was released in April 1987 as the second single from his album Americana Michael Martin Murphey album Americana . The song reached number one on the U.S. and Canadian country charts in August 1987. Chart performance class wikitable align left Chart 1987 align center Peak br position align left U.S. Billboard Hot Country Singles align center 1 align left Canadian RPM Country Tracks Chart align center 1 start box s bef rows 2 before One Promise Too Late BR by Reba McEntire s ttl title Billboard magazine Billboard Hot Country Songs Hot Country Singles br number one single years August 15, 1987 s aft after Why Does It Have to Be Wrong or Right BR by Restless Heart s ttl title RPM magazine RPM Country Tracks br number one single years August 22 August 29, 1987 s aft after Whiskey, If You Were a Woman BR by Highway 101 end box References Jurek, Thom. Allmusic class album id r93268 pure url yes Americana , Allmusic . DEFAULTSORT LongLine of Love, The Category Michael Martin Murphey songs Category Songs written by Paul Overstreet Category 1987 singles Category Billboard Hot Country Songs number one singles Category RPM Country Tracks number one singles Category Songs written by Thom Schuyler Category Songs produced by Jim Ed Norman 1980s ... more details
Infobox Album See Wikipedia WikiProject Albums Name LongLine of Leavers Type Album Artist Caedmon s Call Cover CaedmonsCall LongLineOfLeavers.jpg Released 10 October 2000 Recorded Franklin, TN Genre Rock music , Christian rock Length 54 59 Label Essential Records Christian Essential Records Producer Ed Cash & Caedmon s Call mostly Reviews Allmusic Rating 4 5 Allmusic class album id r501298 pure url yes link Last album 40 Acres br 1999 This album LongLine of Leavers br 2000 Next album Songs from the Guild br 2000 LongLine of Leavers is a studio album from Caedmon s Call . It was recorded at the following studios The Bennett House Franklin, TN Engineer Ben Wisch The Farm Marvin, NC Engineer Ed Cash Sound Kitchen Franklin, TN Engineer Jim Dineen Produced by Monroe Jones NOTOC Track listing Only One 3 47 Love Is Different 4 33 Prepare Ye the Way 4 30 Prove Me Wrong 2 50 Mistake of My Life 3 53 Masquerade 4 52 What You Want 4 03 Valleys Fill First 4 12 Can t Lose You 4 59 Love Alone 4 55 Dance 5 29 Piece of Glass 4 09 Ballad of San Francisco 2 50 Personnel Band members Cliff Young Lead vocals, Acoustic Rhythm guitar, Backing Vocals Danielle Young Vocals, Backing Vocals Garett Buell Percussion, Drums Jeff Miller Bass Guitar Todd Bragg Drums, Percussion Joshua Moore Hammond B 3, Piano, Rhodes, Wurlitzer, Accordion, Harmonica Derek Webb Acoustic Guitar, Electric Guitar, Banjo, Vocals, Backing Vocals Guest musicians Byron House Upright Bass Ed Cash Acoustic Guitars, Whistle, Backing Vocals, Mandolin, Rhodes Swells Barry Green Trombone George Cocchini Electric Guitar Mike Haynes Trumpet David Angall strings John Catchings strings David Davidson strings Conni Ellisor strings Carl Gorodetsky strings Richard Grosjean strings Anthony LaMarchina strings Pamela Sixfin strings Mary Kathryn Vanosdale strings Kristin Wilkinson strings Aaron Senseman Electric Guitar Gary Burnette Electric Guitar Mark Hill Bass Jeff Roach Keyboards Monroe Jones Keyboards Release details 2000, USA, Essential ... more details
one surface and one edge. Such shapes are an object of study in topology. Topology from the Greek ... or analysis situs Greek Latin for picking apart of place . This later acquired the modern name of topology Specify . By the middle of the 20th century, topology had become an important area of study within mathematics. The word topology is used both for the mathematical discipline and for a family ... object of topology. Of particular importance are homeomorphism s , which can be defined as continuous function s with a continuous inverse function inverse . Topology includes many subfields. The most basic and traditional division within topology is General topology point set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces basic examples include compactness and connectedness algebraic topology , which generally tries ... mathematics homology and geometric topology , which primarily studies manifold s and their embeddings placements in other manifolds. Some of the most active areas, such as low dimensional topology and graph ... , the simplest non trivial knot See also topology glossary for definitions of some of the terms used in topology and topological space for a more technical treatment of the subject. History Image Konigsberg .... Topology began with the investigation of certain questions in geometry. Leonhard Euler s 1736 ... academic treatises in modern topology. The term Topologie was introduced in German in 1847 by Johann ... years in correspondence before its first appearance in print. Topology, its English form, was first ... topologist in the sense of a specialist in topology was used in 1905 in the magazine The Spectator ... definition of topology. Modern topology depends strongly on the ideas of set theory , developed ... of homotopy and homology mathematics homology , which are now considered part of algebraic topology ..., see point set topology and algebraic topology . Elementary introduction Topology, as a branch ... more details
0, Closed ordinal space 0, Uncountable discrete ordinal space LonglinetopologyLonglineLonglinetopology Extended longline An altered Longlinetopologylongline Lexicographic order topology ... s condensed corkscrew Thomas s plank Thomas s corkscrew Weak parallel linetopology Strong parallel linetopology Concentric circles Appert space Maximal compact topology Minimal Hausdorff topology ...Infobox Book name Counterexamples in Topology image image caption author Lynn Steen Lynn Arthur Steen ... Counterexamples in Topology 1970, 2nd ed. 1978 is a book on mathematics by topology topologist s Lynn ... a counterexample which exhibits one property but not the other. In Counterexamples in Topology , Steen ... , Minnesota in the summer of 1967, canvassed the field of topology for such counterexamples and compiled ... space which is not second countable space second countable is counterexample 3, the discrete topology ... of metrization theory and general topology see History of the separation axioms for more. List of mentioned counterexamples colbegin cols 2 finite set Finite discrete topology Countable discrete topology Uncountable discrete topology Indiscrete topology Partition topology Odd even topology Deleted integer topology Particular point topology Finite particular point topology Particular point topology Countable particular point topology Particular point topology Uncountable particular point topology Sierpinski space , see also particular point topology Closed extension topology Finite excluded point topology Countable excluded point topology Uncountable excluded point topology Open extension topology Either or topology Finite complement topology on a countable space Finite complement topology on an uncountable space Countable complement topology Double pointed countable complement topology Compact complement topology Countable Fort space Uncountable Fort space Fortissimo space Arens Fort space Modified Fort space Euclidean space Euclidean topology Cantor set Rational number s Irrational ... more details
The White Line was a short lived branch of the Long Island Rail Road in western Queens County, New York . Officially known as the Newtown and Flushing Railroad , the line was chartered in 1871, but was only in service from 1873 1876. History In the first two years of its existence, the Long Island Rail Road enjoyed a monopoly of rail service on Long Island. This came to an end in the early 1850 s when competition came in the form of the Flushing Railroad , which ran along the north shore of Queens County, New York . In 1859, the FRR became the New York and Flushing Railroad , but as they attempted ... to the bright white passenger cars used on the line, it was nicknamed the White Line. The service began at the original Hunter s Point station, today s Long Island City LIRR station and ran along the main line through Woodside Station. Southeast of Winfield Junction LIRR station Winfield Junction it split off between the Main LineLong Island Rail Road and what is today the Port Washington Branch . Running on a separate right of way south of the Port Washington Branch and north of the Main Line, it had its own Newtown Station, then the line curved northeast where it had a station for Corona ... Railroad of Long Island branched off to head towards Garden City, Bethpage and Babylon. ref http ... the Flushing and North Side and the Central Railroad of Long Island were consolidated into the Flushing ..., the LIRR no longer saw any need for the White Line and it was abandoned. Flushing Village Railroad Station list Hunter s Point LIRR station Schwalenberg s Park LIRR station Sunnyside LIRR White Line station Woodside LIRR station Winfield LIRR station Newtown LIRR White Line station Corona Park LIRR ... The Newtown and Flushing Railroad The White Line Arrt s Arrchives http www.lirrhistory.com lirrsta.html Earliest Stations of the LIRR Unofficial LIRR History Long Island Rail Road Category Long Island Rail Road Category Transportation in Queens Category Railway lines opened in 1873 Category ... more details
Infobox local1 name World s First br Long Distance Telephone Line chisl yes image caption location Pleasant Valley Road br French Corral , California lat degrees lat minutes lat seconds lat direction long degrees long minutes long seconds long direction locmapin California built 1877 architect Ridge Telephone Company architecture added 1937 03 25 designated chisl number 247 governing body refnum The world s first long distance telephone line , established in 1877, connected French Corral with Bowman Lake California Bowman Lake previously known as French Lake ref cite book url http books.google.com books?id YYj5akx5i5sC&pg PA149&lpg PA149&dq 22World 27s First Long Distance Telephone Line 22&source web&ots 0otEQZJQhQ&sig rAjjPPqwAS3AY474dUAyOPW7RNI&hl en&sa X&oi book result&resnum 10&ct result pages 149 title Traveling California s gold rush country last Kelly first Leslie A. year 1997 publisher Falcon location Helena, Montana isbn 1560444843 ref at the headwaters of the Yuba River . ref cite web url http tedmuller.us Outdoor Hiking 2007 070625 MountLolaWhiteRock.htm title Bowman Lake, headwaters ... Landmark. The marker is inscribed blockquote WORLD S FIRST LONG DISTANCE TELEPHONE LINE br br The first long distance telephone in the world, built in 1877 by the Ridge Telephone Company, connected French ... Detail.aspx?num 247 title WORLD S FIRST LONG DISTANCE TELEPHONE LINE accessdate 2008 08 03 ... books?id AYMPR6xAj50C&printsec frontcover PPA260,M1 ref The line was operated by the Ridge Telephone Company for the service of Milton Mining and Water Company, as well as other water companies. The line ... http books.google.com books?id mtAoDbEEYQkC&pg PA198&lpg PA198&dq 22World 27s First Long Distance Telephone Line 22&source web&ots Qw96n4KxH &sig asTaYyk8WPGxqAy0p Sh3jE3vzE&hl en&sa X&oi book result&resnum ... History of the telephone Timeline of the telephone California history DEFAULTSORT First Long Distance Telephone Line Category 1877 establishments Category History of Nevada County, California Category ... more details
Other uses2 Main Line Infobox rail line name LIRR Main Line color LIRR color City Main type Commuter rail system Long Island Rail Road image Grandavlirrmain.JPG status Operational locale Long Island , New York , United States USA start Long Island City LIRR station Long Island City end Greenport LIRR station Greenport stations 29 passenger, 1 employee only routes Rail color box system LIRR line City Main rail color box system LIRR line Belmont Park Rail color box system LIRR line Hempstead Rail color box system LIRR line Oyster Bay Rail color box system LIRR line Port Jefferson Rail color box system LIRR line Ronkonkoma ridership open owner Long Island Rail Road operator Long Island Rail Road ... from Long Island City most Main Line trains use those tunnels rather than running to or from Long Island City. Continuing east, five branches split from the Main Line in order from west to east, they are Port ..., connecting the Main Line to the Montauk Branch Public timetables refer to the Main Line east of Hicksville as the Ronkonkoma Branch . Trains on the Main Line between Long Island City and Ronkonkoma ... description and current train service The Main Line has two tracks from just east of Long Island ... Main Line track To accommodate an expected increase in Long Island Rail Road ridership once the East .... Montauk Branch trains that use the Main Line and Central Branch Long Island Rail Road Central Branch ... LineLong Island Rail Road reflist Long Island Rail Road Category Long Island Rail Road Category Transportation ... br west of Ronkonkoma LIRR station Ronkonkoma speed elevation map The Main Line is a Track rail transport rail line owned and operated by the Long Island Rail Road in the U.S. state of New York . It begins in Long Island City and runs along the middle of Long Island about 95 miles 153 km to Greenport, Suffolk County, New York Greenport . A mile east of Long Island City east of Hunterspoint ... LIRR station Hicksville station Central Branch Long Island Rail Road Central Branch at Beth Interlocking ... more details
. See also lower limit topologylonglinetopology Linear continuum Notes references References ... in the real line. The same argument as before shows that the subspace topology on Z is not equal to the induced ...one source date October 2011 In mathematics , an order topology is a certain topology that can be defined on any totally ordered set . It is a natural generalization of the topology of the real numbers ... generate nice non Hausdorff counter examples If X is a totally ordered set, the order topology ... a,b x mid a x b math together with the above rays form a base topology base for the order topology ... intervals and rays. The order topology makes X into a completely normal space completely normal ... topology If Y is a subset of X , then Y inherits a total order from X . Y therefore has an order topology, the induced order topology . As a subset of X , Y also has a subspace topology . The subspace topology is always at least as finer topology fine as the induced order topology, but they are not in general ... numbers rationals . Under the subspace topology, the singleton set 1 is open in Y , but under the induced order topology, any open set containing 1 must contain all but finitely many members of the space. An example of a subspace of a linearly ordered space whose topology is not an order topology Though the subspace topology of Y 1 &cup 1 n sub n &isin N sub in the section above is shown to be not generated by the induced order on Y , it is nonetheless an order topology on Y indeed, in the subspace topology every point is isolated i.e., singleton y is open in Y for every y in Y , so the subspace topology is the discrete topology on Y the topology in which every subset of Y is an open set , and the discrete topology on any set is an order topology. To define a total order on Y that generates the discrete topology on Y , simply modify the induced order on Y by defining 1 to be the greatest ... call it say sub 1 sub we have 1 n sub 1 sub 1 for all n &isin N . Then, in the order topology on Y ... more details
This is a list of useful examples in general topology , a field of mathematics . Alexandrov topology Cantor space Co kappa topology Cocountable topology Cofinite topology Compact open topology Compactification mathematics Compactification Discrete space Discrete topology Double pointed cofinite topology Extended real number line Finite topological space Hawaiian earring Hilbert cube Irrational cable on a torus Lakes of Wada LonglinetopologyLongline Order topology Lexicographical order Lexicographical dictionary order Ordinal number Ordinal number topology Real line Split interval Overlapping interval topology Moore plane Sierpinski space Lower limit topology Sorgenfrey line Sorgenfrey plane Space filling curve Topologist s sine curve Trivial topology Unit interval Zariski topology See also Counterexamples in Topology Category Mathematics related lists General topology Category General topology Category Topological spaces ... more details
Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation stretching without tearing or gluing these properties are the topological invariants. Topology may also refer to Topology, the collection of open sets used to define a topological space Topology journal Topology journal , a mathematical journal, with an emphasis on subject areas related to topology and geometry Topology, a term used in architecture to describe spatial effects which cannot be described by topography, i.e., social, economical, spatial or phenomenological interactions Topology, a term used in cell biology to describe the Membrane topology specific orientation of transmembrane proteins . Topology electronics , a configuration of electronic components. Network topology , a term used to describe configurations of computer or biological networks. Topology musical ensemble , an Australian post classical quintet Geospatial topology is the study or science of places with applications in earth science , geography , human geography , and geomorphology . In geographic information system s and their data structures, the terms Geospatial topologytopology and planar enforcement are used to indicate that the border line between two neighboring areas and the border point between two connecting lines is stored only once. Thus, any rounding errors might move the border, but will not lead to gaps or overlaps between the areas. Also in cartography, a topological map is a much simplified map that preserves the mathematical topology while sacrificing scale and shape Topology is often confused with the geographic meaning of topography originally the study of places . The confusion may be a factor in topographies having become confused with terrain or relief , such that they are essentially synonymous. In phylogenetics , the branching pattern of a phylogenetic tree. TopologiLinux , a Linux distribution disambig bar Topologie de Topologie es Topolog a desambiguaci n ... more details
. History General topology grew out of a number of areas, most importantly the following the detailed study of subsets of the real line once known as the topology of point sets , this usage is now obsolete ...incomplete date August 2009 In mathematics , general topology or point set topology is the branch of topology ... from other branches of topology in that the topological spaces may be very general, and do not have to be at all similar to manifold s. General topology provides the most general framework where fundamental concepts of topology such as open closed sets, continuity, interior exterior boundary points, and limit points could be defined. Definition Main Topological Space A topology is a pair X , ... s, in the early days of functional analysis . General topology assumed its present form around 1940 ..., it is in general topology that basic notions are defined and theorems about them proved. This includes the following open set open and closed set s interior topology interior and closure topology closure neighbourhood topology neighbourhood and closeness topology closeness compact space compactness and connected space connectedness continuous function topology continuous function mathematics ... branches of mathematics. Set theoretic topology examines such questions when they have substantial relations to set theory , as is often the case. Other main branches of topology are algebraic topology , geometric topology , and differential topology . As the name implies, general topology provides the common foundation for these areas. An important variant of general topology is pointless topology , which, rather than using sets of points as its foundation, builds up topological concepts ... and locales . See also List of examples in general topology Glossary of general topology for detailed definitions List of general topology topics for related articles Category of topological spaces References Some standard books on general topology include Bourbaki cite Topologie G n rale cite ... more details
In mathematics , the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton mathematics singleton math a math is the order section math a x leq a math for each math a in X math . If math leq math is a partial order, the upper topology is the least order consistent topology in which the open set s are the up set s. The lower topology induced by the preorder is defined similarly in terms of the down set s. The preoder inducing the upper topology is its specialization preorder , but the specialization preorder of the lower topology is opposite to the inducing preorder. The real upper topology is most naturally defined on the upper extended real line math infty, infty mathbb R cup infty math by the system math a, infty a in mathbb R cup pm infty math of open sets. Similarly, the real lower topology math infty,a a in mathbb R cup pm infty math is naturally defined on the lower real line math infty, infty mathbb R cup infty math . A real function on a topological space is upper semi continuous if and only if it is lower continuous, i.e. is Continuous function continuous with respect to the lower topology on the lower extended line math infty, infty math . Similarly, a function into the upper real line is lower semi continuous if and only if it is upper continuous, i.e. is Continuous function continuous with respect to the upper topology on math infty, infty math . References cite book author Gerhard Gierz coauthors K.H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott title Continuous Lattices and Domains publisher Cambridge University Press date 2003 isbn 0 521 80338 1 page 510 cite book last Kelley first John L. authorlink John L. Kelley title General Topology publisher Van Nostrand Reinhold date 1955 page 101 cite book last Knapp first Anthony W. title Basic Real Analysis publisher Birkhhauser date 2005 isbn 0817632506 page 481 Category General topology Category Order theory topology stub ... more details
Image with unknown copyright status removed Image line network.gif frame Image showing line network layout A linear bus topology is a network topology consisting of a main run of cable with a terminator at each end. All nodes file server, workstations, and peripherals are connected to the linear cable. Ethernet and LocalTalk networks use a linear bus topology. Advantages of a linear bus topology Easy to connect a computer or peripheral to a linear bus. Requires less cable length than a star topology . Disadvantages of a linear bus topology Entire network shuts down if there is a break in the main cable. Terminators are required at both ends of the backbone cable. Difficult to identify the problem if the entire network shuts down. Not meant to be used as a stand alone solution in a large building. External links http fcit.usf.edu network chap5 chap5.htm Category Network topology compu network stub id Topologi runtut ... more details
for the mathematical journal Geometry & Topology In mathematics , geometry and topology is an umbrella term for geometry and topology , as the line between these two is often blurred, most visibly in Riemannian ... like the Gauss Bonnet theorem and Chern Weil theory . Sharp distinctions between geometry and topology can be drawn, however, as discussed below. It is also the title of a journal Geometry & Topology that covers these topics. Scope It is distinct from geometric topology , which more narrowly involves applications of topology to geometry. It includes Differential geometry and topology Geometric topology including low dimensional topology and surgery theory It does not include such parts of algebraic topology as homotopy theory , but some areas of geometry and topology such as surgery theory, particularly algebraic surgery theory are heavily algebraic. Distinction between geometry and topology Pithily, geometry has local structure or infinitesimal , while topology only has global structure. Alternatively, geometry has continuous moduli, while topology has discrete moduli. By examples, an example of geometry is Riemannian geometry , while an example of topology is homotopy theory . The study of metric space s is geometry, the study of topological space s is topology. The terms are not used ... beyond dimension . So differentiable structures on a manifold is an example of topology. By contrast ... structure is topology. If have non trivial deformations, the structure is said to be flexible , and its ... so studying maps up to homotopy is topology. Similarly, differentiable structures on a manifold is usually a discrete space, and hence an example of topology, but exotic R4 exotic R sup 4 sup s have ... symplectic topology and symplectic geometry . By Darboux s theorem , a symplectic manifold has no local structure, which suggests that their study be called topology. By contrast, the space of symplectic ... Geometry And Topology Category Topology Category Geometry ... more details
In mathematics, and especially general topology , the Euclidean topology is an example of a topology given to the set of real number s, denoted by R . To give the set R a topology means to say which subset s of R are open , and to do so in a way that the following axiom s are met ref name CEIT Citation first L. A. last Steen first2 J. A. last2 Seebach title Counterexamples in Topology publisher Dover year 1995 ISBN 048668735X ref The union mathematics union of open sets is an open set. The finite intersection mathematics intersection of open sets is an open set. The set R and the empty set are open sets. Construction The set R and the empty set are required to be open sets, and so we define R and to be open sets in this topology. Given two real numbers, say x and y , with nowrap 1 x y we define an uncountably infinite family of open sets denoted by S sub x , y sub as follows ref name CEIT math S x,y r in bold R x r y . math Along with the set R and the empty set , the sets S sub x , y sub with nowrap 1 x y are used as a basis topology basis for the Euclidean topology. In other words, the open sets of the Euclidean topology are given by the set R , the empty set and the unions and finite intersections of various sets S sub x , y sub for different pairs of x , y . Properties The real line, with this topology, is a T5 space T sub 5 sub space . Given two subsets, say A and B , of R with nowrap 1 font style text decoration overline A font B A font style text decoration overline B font , where font style text decoration overline A font denotes the closure topology closure of A , etc., there exist open sets S sub A sub and S sub B sub with nowrap 1 A S sub A sub and nowrap 1 B S sub B sub such that nowrap 1 S sub A sub S sub B sub . ref name CEIT References Reflist Category Topology es Topolog a euclideana nl Euclidische topologie ... more details
In topology and related areas of mathematics , a product space is the cartesian product of a family of topological space s equipped with a natural topology called the product topology . This topology differs from another, perhaps more obvious, topology called the box topology , which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is correct in that it makes the product space a product category theory categorical product of its factors, whereas the box topology is Comparison of topologies too fine this is the sense in which the product topology is natural . Definition Given X such that math ... p sub i sub X &rarr X sub i sub , the product topology on X is defined to be the coarsest topology i.e. the topology with the fewest open sets for which all the projections p sub i sub are continuous topology continuous . The product topology is sometimes called the Tychonoff topology . The open sets in the product topology are unions finite or infinite of sets of the form math prod U i ... many times. The product topology on X is the topology generated by sets of the form p sub i sub ... p sub i sub sup &minus 1 sup U form a subbase for the topology on X . A subset of X is open if and only ... a basis topology basis for the product topology using bases of the constituting spaces X sub i sub . A basis ... topology on X . In general, the box topology is finer topology finer than the product topology, but for finite products they coincide. Examples If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on R sup n sup . The Cantor set is homeomorphic to the product of countable countably ... topology. Several additional examples are given in the article on the initial topology . Properties ... if W is a subspace topology subspace of the product space whose projections down to all the X sub ... more details
Long Melford Bury St Edmunds branch line The Long Melford Bury St Edmunds branch line was a railway between Long Melford on the Stour Valley Railway and Bury St Edmunds on the Ipswich to Ely Line . The line opened on 9 August 1865 and closed to passengers on 10 April 1961 and freight on 19 April 1965. Sources http www.subbrit.org.uk sb sites stations l lavenham index.shtml Bury St Edmunds Long Melford branch at Subterranea Britannica Category Rail transport in Suffolk Category Closed railway lines in the East of England Category Railway lines opened in 1865 Category Railway lines closed in 1965 England rail transport stub ... more details
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