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92-628: The Tube map (sometimes called the London Underground map ) is a schematic transport map of the lines, stations and services of the London Underground , known colloquially as "the Tube", hence the map's name. The first schematic Tube map was designed by Harry Beck in 1931. Since then, it has been expanded to include more of London's public transport systems , including the Docklands Light Railway , London Overground ,

184-416: A robot can be described by a manifold called configuration space . In the area of motion planning , one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose. Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components. In order to create a continuous join of pieces in

276-497: A smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting

368-412: A convenient proof that any subgroup of a free group is again a free group. Differential topology is the field dealing with differentiable functions on differentiable manifolds . It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require only

460-426: A given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected. Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric . In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x

552-420: A homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to

644-573: A limited service has declined in recent years, as patronage has recovered from its early-1980s low. As there are now fewer restrictions to show, most of the remaining ones are now indicated in the accompanying text, rather than by special line markings. The Tube map exists to help passengers navigate the London rapid transit network, and whether it should play a wider role in helping people navigate London itself has been questioned. The question has been raised as to whether mainline railways should be shown on

736-462: A series of London Buses " spider diagrams " to display at bus stops around the city, conveying bus route information in a schematic style similar to Beck's design, with straight lines and 45° angles depicting geographically distorted bus routes, coloured lines and numbers to differentiate services, and graphical markers to show bus stops. Tube and rail lines are not included, but interchanges are denoted with appropriate symbols by bus stop names, such as

828-534: A set (for instance, determining if a cloud of points is spherical or toroidal ). The main method used by topological data analysis is to: Several branches of programming language semantics , such as domain theory , are formalized using topology. In this context, Steve Vickers , building on work by Samson Abramsky and Michael B. Smyth , characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties. Topology

920-542: A significant number of pages to schematic diagrams. Topology Topology (from the Greek words τόπος , 'place, location', and λόγος , 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations , such as stretching , twisting , crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space

1012-407: A televised search for the most well-known British design icon. It is widely cited by academics and designers as a 'design classic', and those cultural associations make London Underground not usually permit the design to be used or altered for any other purpose. That has been officially sanctioned only on a few occasions: Stylistic aspects of the London diagram, such as the line colours and styles and

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1104-457: Is a π -system . The members of τ are called open sets in X . A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set ), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x . A function or map from one topological space to another

1196-409: Is a set endowed with a structure, called a topology , which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity . Euclidean spaces , and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies . A property that

1288-665: Is a compromise between a purely abstract diagram (e.g. the schematic of the Washington Metro) and an exclusively realistic representation (e.g. the corresponding aerial view of Washington). In electrical and electronic industry, a schematic diagram is often used to describe the design of equipment. Schematic diagrams are often used for the maintenance and repair of electronic and electromechanical systems. While schematics were traditionally drawn by hand, using standardized templates or pre-printed adhesive symbols, today electronic design automation software (EDA or "electrical CAD")

1380-661: Is a quantum field theory that computes topological invariants . Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory , the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson , Jones , Witten , and Kontsevich have all won Fields Medals for work related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in string theory , as different manifolds can sustain different kinds of strings. In cosmology, topology can be used to describe

1472-406: Is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus . If a continuous function is one-to-one and onto , and if the inverse of the function is also continuous, then the function is called

1564-444: Is invariant under such deformations is a topological property . The following are basic examples of topological properties: the dimension , which allows distinguishing between a line and a surface ; compactness , which allows distinguishing between a line and a circle; connectedness , which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Wilhelm Leibniz , who in

1656-782: Is often used. In electronic design automation, until the 1980s schematics were virtually the only formal representation for circuits. More recently, with the progress of computer technology, other representations were introduced and specialized computer languages were developed, since with the explosive growth of the complexity of electronic circuits, traditional schematics are becoming less practical. For example, hardware description languages are indispensable for modern digital circuit design. Schematics for electronic circuits are prepared by designers using EDA ( electronic design automation ) tools called schematic capture tools or schematic entry tools. These tools go beyond simple drawing of devices and connections. Usually they are integrated into

1748-776: Is point-set topology. The basic object of study is topological spaces , which are sets equipped with a topology , that is, a family of subsets , called open sets , which is closed under finite intersections and (finite or infinite) unions . The fundamental concepts of topology, such as continuity , compactness , and connectedness , can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby , arbitrarily small , and far apart can all be made precise by using open sets. Several topologies can be defined on

1840-453: Is relevant to physics in areas such as condensed matter physics , quantum field theory and physical cosmology . The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science . Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials. The compressive strength of crumpled topologies

1932-464: Is studied in attempts to understand the high strength to weight of such structures that are mostly empty space. Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics. A topological quantum field theory (or topological field theory or TQFT)

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2024-437: Is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside. In one of the first papers in topology, Leonhard Euler demonstrated that it

2116-428: Is the set of all points whose distance to x is less than r . Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line , the complex plane , real and complex vector spaces and Euclidean spaces . Having a metric simplifies many proofs. Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal

2208-437: Is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. The most important of these invariants are homotopy groups , homology, and cohomology . Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for

2300-612: The DLR station at City Airport ) are shown with a black aeroplane symbol. Since 2000, stations with a nearby interchange to river bus piers on the Thames have been marked with a small boat symbol to promote London River Services . When Eurostar services used Waterloo International , the Eurostar logo was shown next to Waterloo station . In November 2007, the terminus was transferred to St Pancras International . The Tube map aims to make

2392-571: The Elizabeth line , Tramlink , the London Cable Car and Thameslink . As a schematic diagram, it shows not the geographic locations but the relative positions of the stations , lines , the stations' connective relations and fare zones . The basic design concepts have been widely adopted for other such maps around the world and for maps of other sorts of transport networks and even conceptual schematics. A regularly updated version of

2484-529: The Northern line . In 1997, Beck's importance was posthumously recognised, and as of 2022, this statement is printed on every Tube map: "This diagram is an evolution of the original design conceived in 1931 by Harry Beck". By 1960, Beck had fallen out with the Underground's publicity officer, Harold Hutchison, who was not a designer himself but drafted his own version of the Tube map that year. It removed

2576-429: The geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries. 2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from

2668-465: The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick ." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg , the result does not depend on

2760-599: The plane , the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane , which cannot (that is, all their realizations are surfaces that are not manifolds). General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology

2852-505: The real line , the complex plane , and the Cantor set can be thought of as the same set with different topologies. Formally, let X be a set and let τ be a family of subsets of X . Then τ is called a topology on X if: If τ is a topology on X , then the pair ( X , τ ) is called a topological space. The notation X τ may be used to denote a set X endowed with the particular topology τ . By definition, every topology

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2944-443: The 17th century envisioned the geometria situs and analysis situs . 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. The motivating insight behind topology

3036-463: The British designer Mark Noad unveiled his vision for a more 'geographically accurate' London Underground map. The map is an attempt to see if it is possible to create a geographically accurate representation of the Underground system and still retain some of the clarity of Beck's original diagram. It uses similar principles, fixed-line angles (30° and 60°, instead of 45°) and shortens the extremities of

3128-574: The District and Piccadilly lines were included for the first time in 1933 with Harry Beck 's first proper Tube map, the portion of the Metropolitan line beyond Rickmansworth did not appear until 1938, and the eastern end of the District line did not appear until the mid-1950s. The route map continued to be developed and was issued in various formats and artistic styles until 1920, when, for

3220-709: The London Cable Car and the boundaries of fare zones. Some commentators have suggested that Beck's design should be replaced with a new design that can incorporate the new lines more comfortably. Since 2004, Art on the Underground has commissioned various British and international artists to create a cover for the pocket map. These free maps are one of the largest public art commissions in the UK, with millions of copies printed. Over 35 different designs have been produced, from artists such as Rachel Whiteread , Yayoi Kusama , Tracey Emin and Daniel Buren . The designers of

3312-469: The London Underground map (including 45° angles, evenly-spaced 'stations' and some geographic distortion) has been emulated by many other underground railway systems around the world. While London Underground have been protective of their copyright they have also allowed their concepts to be shared with other transport operators ( Amsterdam 's GVB even pays tribute on its map). The success of

3404-473: The Tube map as a piece of information design has led to many imitations of its format. What is probably the earliest example is the Sydney Suburban and City Underground railway map of 1939. It follows Beck's styling cues, and in size, design and layout, it is nearly a clone of the London map of the late 1930s, right down to the use of the Underground roundel . In 2002, Transport for London launched

3496-583: The Tube roundel. Unlike the traditional Tube map, the bus maps display services appropriate to specific transport hubs rather than a full network. Each map also contains a central rectangle of a simple geographically accurate street map to display the positions of bus stops; outside the rectangle, the only geographic feature to appear on the bus maps is the River Thames. The maps are also available for electronic download, with map collections ordered by London borough councils. The bus maps were designed for TfL by

3588-445: The Tube stops that connect with National Rail services, links to airports and River Services . In some cases, stations within short walking distance are now shown, often with the distance between them, such as Fenchurch Street 's distance from Tower Hill (an evolution of the pedestrian route between Bank and Monument stations , which was once prominently marked on the map). Further, step-free access notations are also incorporated in

3680-456: The UERL and one from each of the other four companies: A geographical map presented restrictions since for sufficient clarity of detail in the crowded central area of the map required the extremities of the District and Metropolitan lines to be omitted and so a full network diagram was not provided. The problem of truncation remained for nearly half a century. Although all of the western branches of

3772-485: The addition of the South London line to London Overground was supposed to cause the southern loop to be added to future Tube maps in late 2010, and, as of May 2013, it is up and running. Like many other rapid transit maps, because the Tube map ignores geography, it may not accurately depict the relative orientation and distance between stations. Transport for London formerly published several bus maps that depicted

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3864-492: The approximate paths of tube routes relative to major streets and London bus routes . The maps also show locations of certain cultural attractions and geographic landmarks. Internet mapping services such as Google Maps offer a "Transit Layer" showing actual routes superimposed on the standard street map. A map shows Underground, London Overground, Elizabeth line, DLR lines and National Rail stations within Zone 1–2. The 'look' of

3956-460: The cartographic design company T-Kartor group. An isochrone map of the network was made available in 2007. In 2009, British Waterways produced a map of London's waterways in a Tube-style diagrammatic map, depicting the River Thames , the various canals and subterranean rivers in the city. Attempts to create alternative versions to the official Tube map have continued. In June 2011,

4048-466: The complicated network of services easy to understand, but it is not possible to have complete information about the services that operate on each line. Limited-service routes have sometimes been identified with hatched lines, with some complications added to the map to show where peak-only services ran through to branches such as that to Chesham on the Metropolitan line. The number of routes with

4140-509: The concepts now known as homotopy and homology , which are now considered part of algebraic topology . Unifying the work on function spaces of Georg Cantor , Vito Volterra , Cesare Arzelà , Jacques Hadamard , Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined

4232-555: The corresponding line. In 1964, the design of the map was taken over by Paul Garbutt, who, like Beck, had produced a map in his spare time because of his dislike of the Hutchison design. Garbutt's map restored curves and bends to the diagram but retained Hutchison's black interchange circles, although squares were replaced with circles with a dot inside. Garbutt continued to produce Underground maps for at least another 20 years. Tube maps stopped bearing their designer's name in 1986, when

4324-482: The current map. In addition, the fare zones have been added to help passengers judge the cost of a journey. One of the major changes to be made to the revision of the Tube map put out in September 2009 was the removal of the River Thames . The river had been omitted from official maps on several previous occasions (for example, according to David Leboff and Tim Demuth's book, in 1907, 1908 and 1919), and from 1921 it

4416-458: The definition of sheaves on those categories, and with that the definition of general cohomology theories. Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on

4508-577: The design, such as changing the interchange symbol from a diamond to a circle and altering the line colours of the Central line from orange to red and of the Bakerloo line from red to brown. Beck's final design, in 1960, bears a strong resemblance to the current map. Beck lived in Finchley , North London , and one of his maps is still preserved on the southbound platform at Finchley Central station , on

4600-494: The details that would be repeated on each phase of a three-phase system, showing only one element instead of three. Electrical diagrams for switchgear often have common device functions designate by standard function numbers . Another type of diagram used for power systems is a three-line diagram . For analysis purposes of a power system, from the one-line diagram, if the system is balanced, an equivalent per-phase (or single-phase ) schematic diagram can be obtained. If all of

4692-643: The doughnut. While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n -dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n . Lines and circles , but not figure eights , are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces , although not all surfaces are manifolds. Examples include

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4784-498: The elements of the map then had a very strong resemblance to today's map. While the standard Tube map avoided representing most mainline services, a new variant of the map issued in 1973, the "London's Railways" map, was the first to depict Tube and above-ground mainline rail services in a diagrammatic style that closely matched Beck's designs. That version was created by Tim Demuth of the London Transport publicity office and

4876-518: The expense of considerably increased complexity, as they contain almost 700 stations. Some non-Underground lines have appeared on the standard tube map: When Transport for London expanded its London Overground service to include the East London line in 2010, the East London line, extended to Croydon, changed from a solid orange line to a double orange stripe. According to 2007 proposals,

4968-457: The first time, the geographic background detail was omitted in a map designed by MacDonald Gill . That freed the design to enable greater flexibility in the positioning of lines and stations. The routes became more stylised but the arrangement remained, largely, geographic in nature. The 1932 edition was the last geographic map to be published before Beck's diagrammatic map was introduced. The first diagrammatic map of London's rapid transit network

5060-587: The following edition of the diagram in December 2009 reinstated both the river and fare zones. In more recent years, TfL has expanded its rail services, notably with the expansion of the London Overground network, which has taken over a number of National Rail lines and brought them into the TfL network, each of them being converted lines being added to the Tube map. Further additions have been made such as

5152-407: The hairy ball theorem applies to any space homeomorphic to a sphere. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking

5244-402: The hole into a handle. Homeomorphism can be considered the most basic topological equivalence . Another is homotopy equivalence . This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object. Topology, as a well-defined mathematical discipline, originates in the early part of

5336-506: The inspiration for Beck's map. His colleagues pointed out the similarities, however, and he once produced a joke map with the stations replaced by electrical circuit symbols and names, with terminology such as " Bakerlite " for the Bakerloo line . To that end, Beck devised a simplified map with stations, straight-line segments connecting them, and the River Thames ; and lines running only vertically, horizontally, or on 45° diagonals. To make

5428-532: The lines to make it more compact. In 2013, Dr Max Roberts, a psychology lecturer at the University of Essex with a particular interest in usability , information design and schematic mapping, issued his own version of the Tube map. His design, based on a series of concentric circles, emphasised the concept of the newly completed orbital loop surrounding Central London with radial lines. A map created to illustrate Tube-related articles on Misplaced Pages in 2014

5520-465: The main line station has a different name from the Underground station that it connects with, since 1977 this has been shown in a box. The distance between the Tube station and the main line station is now shown. Contemporary maps have marked stations offering step-free access with a blue circle containing a wheelchair symbol in white. Stations with links to airports ( Heathrow Terminals 2 & 3 , Terminal 4 , and Terminal 5 for Heathrow Airport and

5612-451: The map clearer and to emphasise connections, Beck differentiated between ordinary stations, marked with tick marks, and interchange stations , marked with diamonds. London Underground was initially sceptical of his proposal since it was an uncommissioned spare-time project and was tentatively introduced to the public in a small pamphlet in 1933. However, it immediately became popular, and the Underground has used topological maps to illustrate

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5704-522: The map have tackled a variety of problems in showing information as clearly as possible and have sometimes adopted different solutions. The font for the map, including station names, is Johnston , which uses perfect circles for the letter 'O'. That is historic and the generic font for all TfL uses from station facades to bus destination blinds. The table below shows the changing use of colours since Beck's first map. The current colours are taken from Transport for London's colour standards guide, which defines

5796-479: The map is available from the official Transport for London website. In 2006, the Tube map was voted one of Britain's top 10 design icons which included Concorde , Mini , Supermarine Spitfire , K2 telephone box , World Wide Web and the AEC Routemaster bus. Since 2004, Art on the Underground has been commissioning artists to create covers for the pocket Tube map . As London's early transport system

5888-559: The map, particularly those in Inner London . The Underground has largely resisted adding additional services to the standard Tube map and instead produces separate maps with different information, including: Maps are produced in different sizes, the most common being Quad Royal (40 × 50 inches) poster size and Journey Planner pocket size. The maps showing all the National Rail routes provide useful additional information at

5980-418: The network ever since. Despite the complexity of making the map, Beck was paid just ten guineas for the artwork and design of the card edition (five guineas for the poster). After its initial success, he continued to design the Tube map until 1960, a single (and unpopular) 1939 edition by Hans Scheger being the only exception. Meanwhile, as well as accommodating new lines and stations, Beck continually altered

6072-482: The number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology. Further contributions were made by Augustin-Louis Cauchy , Ludwig Schläfli , Johann Benedict Listing , Bernhard Riemann and Enrico Betti . Listing introduced the term "Topologie" in Vorstudien zur Topologie , written in his native German, in 1847, having used

6164-564: The one-line diagram, three different per-phase schematic diagrams are obtained, known as sequence diagrams : positive sequence diagram , negative sequence diagram , and zero sequence diagram . Each of these diagrams can be represented as an impedance diagram or as an admittance diagram. Schematic diagrams are used extensively in repair manuals to help users understand the interconnections of parts, and to provide graphical instruction to assist in dismantling and rebuilding mechanical assemblies. Many automotive and motorcycle repair manuals devote

6256-475: The overall shape of the universe . This area of research is commonly known as spacetime topology . In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect , and then generalized in other areas of physics, for instance in photonics by F.D.M Haldane . The possible positions of

6348-465: The pairwise arrangement of their intra-chain contacts and chain crossings. Knot theory , a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis . Topological data analysis uses techniques from algebraic topology to determine the large scale structure of

6440-484: The parameters are represented as impedances and voltage sources, the equivalent per-phase schematic diagram is called an impedance diagram . If all of the parameters are represented as admittances and current sources, the equivalent per-phase schematic diagram is called an admittance diagram . If the power system is unbalanced, but it is linear (or can be approximated by a linear system), then Fortescue's theorem ( symmetrical components ) can be applied. In this way, from

6532-481: The planar and higher-dimensional Schönflies theorem . In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory. Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature /spherical, zero curvature/flat, and negative curvature/hyperbolic – and

6624-426: The point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure. Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow

6716-542: The precise colours from the Pantone palette and also a colour naming scheme that is particular to TfL. Earlier maps were limited by the number of colours available that could be clearly distinguished in print. Improvements in colour printing technology have reduced that problem and the map has coped with the identification of new lines without great difficulty. Pecked lines have at various times indicated construction, limited service, or sections closed for renovation. From

6808-423: The schematic aims to capture, on a more general level, the way it works . This may be contrasted with a wiring diagram , which preserves the spatial relationships between each of its components. Schematics and other types of diagrams, e.g., A semi-schematic diagram combines some of the abstraction of a purely schematic diagram with other elements displayed as realistically as possible, for various reasons. It

6900-490: The shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes. To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism . The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and

6992-471: The smoothed corners of Beck's design and created some highly cramped areas (most notably around Liverpool Street station ), and the lines were generally less straight. However, Hutchison also introduced interchange symbols (circles for Underground-only, squares for connections with British Rail main line services) that were black and allowed multiple lines through them, as opposed to Beck, who used one circle for each line at an interchange, coloured according to

7084-408: The space and affecting the curvature or volume. Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology. Some examples of topics in geometric topology are orientability , handle decompositions , local flatness , crumpling and

7176-446: The start, interchange stations were given a special mark to indicate their importance, but its shape has changed over the years. In addition, since 1960, marks were used to identify stations that offered connections with British Rail (now National Rail ). The following shapes have been used: Since 1970 the map has used a reversed (red on white) British Rail "double arrow" beside the station name to indicate main line interchanges. Where

7268-430: The station ticks or interchange symbols, are also frequently used in advertising. Schematic For example, a subway map intended for passengers may represent a subway station with a dot. The dot is not intended to resemble the actual station at all but aims to give the viewer information without unnecessary visual clutter. A schematic diagram of a chemical process uses symbols in place of detailed representations of

7360-651: The term "topological space" and gave the definition for what is now called a Hausdorff space . Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski . Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series . For further developments, see point-set topology and algebraic topology. The 2022 Abel Prize

7452-572: The twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler . His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron . This led to his polyhedron formula , V − E + F = 2 (where V , E , and F respectively indicate

7544-404: The vessels, piping, valves, pumps, and other equipment that compose the system, thus emphasizing the functions of the individual elements and the interconnections among them and suppresses their physical details. In an electronic circuit diagram , the layout of the symbols may not look anything like the circuit as it appears in the physical world: instead of representing the way the circuit looks ,

7636-427: The whole design flow and linked to other EDA tools for verification and simulation of the circuit under design. Programmable logic controllers (PLC) can be programmed using ladder diagrams . In electric power systems design, a schematic drawing called a one-line diagram is frequently used to represent substations , distribution systems or even whole electrical power grids. These diagrams simplify and compress

7728-517: The word for ten years in correspondence before its first appearance in print. The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". Their work was corrected, consolidated and greatly extended by Henri Poincaré . In 1895, he published his ground-breaking paper on Analysis Situs , which introduced

7820-468: Was absent for several years on pocket maps designed by MacDonald Gill . The Thames-free 2009 version was the first time that the river did not appear on the Tube map since the Stingemore pocket map of 1926. The latest removal resulted in widespread international media attention, and general disapproval from most Londoners as well as from the then Mayor of London , Boris Johnson . Based on the reaction,

7912-436: Was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects". The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance,

8004-404: Was designed by Harry Beck in 1931. He was a London Underground employee who realised that because the railway ran mostly underground, the physical locations of the stations were largely irrelevant to the traveller wanting to know how to get from one station to another; only the topology of the route mattered. That approach is similar to that of electrical circuit diagrams although they were not

8096-439: Was impossible to find a route through the town of Königsberg (now Kaliningrad ) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory . Similarly,

8188-597: Was jointly sponsored by British Rail and London Transport. Demuth's map did not replace the standard Tube map but continued to be published as a supplementary resource, later known as the "London Connections" map. Some alterations have been made to the map over the years. More recent designs have incorporated changes to the network, such as the Docklands Light Railway and the extension to the Jubilee line . The map has also been expanded to include routes brought under Transport for London control such as TfL Rail and to note

8280-421: Was operated by a variety of independent companies, there were no complete maps of the network, just for the individual companies' routes. The maps were not typically schematic and were simply the line overlaid on a regular city map . There was no integration of the companies' services or any co-operation in advertising. In 1907, The Evening News commissioned a pocket map, The Evening News London "Tube Map". It

8372-420: Was praised for its clarity and for including future developments such as Crossrail . In July 2015, a map of the network displaying walking calorie burn information for each leg was published by Metro newspaper . The design has become so widely known that it is now instantly recognisable as representing London. It has been featured on T-shirts, postcards and other memorabilia. In 2006, the design came second in

8464-550: Was the first map to show all of the lines with equal weight being given to each line, and it was the first map to use a different colour for each line. Another early combined map was published in 1908 by the Underground Electric Railways Company of London (UERL) in conjunction with four other underground railway companies that used the "Underground" brand as part of a common advertising factor. The map showed eight routes – four operated by

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