topology

• n.

• 形態、位相幾何学、トポロジー

関

form、morphology、topological

　 　 　

WordNet

1. the branch of pure mathematics that deals only with the properties of a figure X that hold for every figure into which X can be transformed with a one-to-one correspondence that is continuous in both directions (同)analysis situs
2. the configuration of a communication network (同)network topology
3. topographic study of a given place (especially the history of the place as indicated by its topography); "Greenlands topology has been shaped by the glaciers of the ice age"
4. develop into a distinctive entity; "our plans began to take shape" (同)take form, take shape, spring
5. assume a form or shape; "the water formed little beads"
6. the visual appearance of something or someone; "the delicate cast of his features" (同)shape, cast
7. a mold for setting concrete; "they built elaborate forms for pouring the foundation"
8. a particular mode in which something is manifested; "his resentment took the form of extreme hostility"
9. a perceptual structure; "the composition presents problems for students of musical form"; "a visual pattern must include not only objects but the spaces between them" (同)shape, pattern
10. (biology) a group of organisms within a species that differ in trivial ways from similar groups; "a new strain of microorganisms" (同)variant, strain, var.
11. the phonological or orthographic sound or appearance of a word that can be used to describe or identify something; "the inflected forms of a word can be represented by a stem and a list of inflections to be attached" (同)word form, signifier, descriptor
12. a printed document with spaces in which to write; "he filled out his tax form"
13. an ability to perform well; "he was at the top of his form"; "the team was off form last night"
14. an arrangement of the elements in a composition or discourse; "the essay was in the form of a dialogue"; "he first sketches the plot in outline form"
15. to compose or represent:"This wall forms the background of the stage setting"; "The branches made a roof"; "This makes a fine introduction" (同)constitute, make
16. create (as an entity); "social groups form everywhere"; "They formed a company" (同)organize, organise
17. the branch of geology that studies the characteristics and configuration and evolution of rocks and land forms (同)geomorphology
18. the admissible arrangement of sounds in words (同)sound structure, syllable structure, word structure
19. studies of the rules for forming admissible words
20. the branch of biology that deals with the structure of animals and plants
21. of or relating to topology (同)topologic
22. the topology of a network whose components are serially connected in such a way that the last component is connected to the first component (同)loop
23. the appearance of the network; "the physical topologies of local area networks include the bus, the ring and the star"
24. the topology of a network whose components are all connected directly to every other component (同)mesh
25. the way the network works; "a network that looks like a star can have the logical topology of a bus"
26. the topology of a network whose components are connected to a hub (同)star

PrepTutorEJDIC

1. 〈C〉〈U〉(色・材質に対して)『形』,形状,姿 / 〈C〉『人体』,体つき / 〈C〉〈U〉『外観』,格好,人影 / 〈C〉〈U〉(ものの現れる)『形態』,形式 / 〈C〉(生物などの)『種類』,品種 / 〈U〉(芸術作品の)表現形式 / 〈U〉(特にスポーツで)(競技者の)フォーム / 〈U〉(心身の)調子,元気 / 〈C〉(印刷された)書き込み用紙,ひな形 / 〈U〉《話》(ある集団などの)やり方,仕方,方式 / 〈C〉〈U〉しきたり,伝統的な儀式 / 〈C〉(英国の学校・米国のある学校で)学年,学級 / 〈U〉〈C〉(意味に対する)形態,形;語形 / 〈C〉(通例寄りかかりのない)長い木のいす / 〈素材〉‘に'形を与える,(…に)…‘を'形作る《+『名』+『into』+『名』》 / (素材から)…‘を'形作る《+『名』+『out of』+『名』》 / …‘を'組織する,構成する / 〈考えなど〉‘を'まとめる,思い付く;〈計画など〉‘を'立てる / 〈習慣など〉‘を'身に付ける;〈交際など〉‘を'結ぶ / 〈文章など〉‘を'組み立てる;〈複数形など〉‘を'作る / (ある順序に)配列する,整列させる / 〈物が〉形を成す / 生じる,現れる / (…に)なる《+『into』+『名』》
2. 形態学(生物の構造の研究) / 形態論,語形論(言語の形態の研究)

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2012/05/24 23:22:59」(JST)

wiki en

[Wiki en表示]

Not to be confused with topography.

For other uses, see Topology (disambiguation).

A Möbius strip, an object with only one surface and one edge. Such shapes are an object of study in topology.

Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.

Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct discipline developed, which was referred to in Latin as the geometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. 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 of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms, which can be defined as continuous functions with a continuous inverse.

Topology includes many subfields. The most basic and traditional division within topology is 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 to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology and graph theory, do not fit neatly in this division. Knot theory studies mathematical knots.

A three-dimensional depiction of a thickened trefoil knot, 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

The Seven Bridges of Königsberg is a famous problem solved by Euler.

Topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first academic treatises in modern topology.

The term "Topologie" was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie, who had used the word for ten years in correspondence before its first appearance in print. "Topology," its English form, was first 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". The term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator.^{[citation needed]} However, none of these uses corresponds exactly to the modern definition of topology.

Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. Cantor, in addition to establishing the basic ideas of set theory, considered point sets in Euclidean space as part of his study of Fourier series.

Henri Poincaré published Analysis Situs in 1895, introducing the concepts of homotopy and homology, which are now considered part of algebraic topology.

Maurice Fréchet, unifying the work on function spaces of Cantor, Volterra, Arzelà, Hadamard, Ascoli, and others, introduced the metric space in 1906. A metric space is now considered a special case of a general topological space. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. In current usage, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.^{[citation needed]}

For further developments, see point-set topology and algebraic topology.

Elementary introduction

Topology, as a branch of mathematics, can be formally defined as "the study of qualitative properties of certain objects (called topological spaces) that are invariant under certain kind of transformations (called continuous maps), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism)." To put it more simply, topology is the study of continuity and connectivity.

The term topology is also used to refer to a structure imposed upon a set X, a structure that essentially 'characterizes' the set X as a topological space by taking proper care of properties such as convergence, connectedness and continuity, upon transformation.

Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics.

The motivating insight behind topology 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.

One of the first papers in topology was the demonstration, by Leonhard Euler, that it 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, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. This problem, the Seven Bridges of Königsberg, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory.

A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and back.

Similarly, 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 the exact shape of the sphere; it applies to pear shapes and in fact 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 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 can't distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. A precise definition of homeomorphic, involving a continuous function with a continuous inverse, is necessarily more technical.

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.

Equivalence classes of the English alphabet:
Homeomorphism	Homotopy equivalence

An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends partially on the font used. The figures use a sans-serif font named Myriad. Notice that homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several of the homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent, e.g. O fits inside P and the tail of the P can be squished to the "hole" part.

Thus, the homeomorphism classes are: one hole two tails, two holes no tail, no holes, one hole no tail, no holes three tails, a bar with four tails (the "bar" on the K is almost too short to see), one hole one tail, and no holes four tails.

The homotopy classes are larger, because the tails can be squished down to a point. The homotopy classes are: one hole, two holes, and no holes.

To be sure we have classified the letters correctly, we not only need to show that two letters in the same class are equivalent, but that two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by suitably selecting points and showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.

Letter topology has some practical relevance in stencil typography. The font Braggadocio, for instance, has stencils that are made of one connected piece of material.

Mathematical definition

Main article: Topological space

Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

1. Both the empty set and X are elements of τ
2. Any union of elements of τ is an element of τ
3. Any intersection of finitely many elements of τ is an element of τ

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.

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 (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always clopen.

A function or map from one topological space to another 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 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. But the circle is not homeomorphic to the doughnut.

Topology topics

Some theorems in general topology

• Every closed interval in R of finite length is compact. More is true: In Rⁿ, a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem).
• Every continuous image of a compact space is compact.
• Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.
• A compact subspace of a Hausdorff space is closed.
• Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
• Every sequence of points in a compact metric space has a convergent subsequence.
• Every interval in R is connected.
• Every compact finite-dimensional manifold can be embedded in some Euclidean space Rⁿ.
• The continuous image of a connected space is connected.
• Every metric space is paracompact and Hausdorff, and thus normal.
• The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
• The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
• Any open subspace of a Baire space is itself a Baire space.
• The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
• On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
• Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.

General topology also has some surprising connections to other areas of mathematics. For example:

• In number theory, Fürstenberg's proof of the infinitude of primes.

See also some counter-intuitive theorems, e.g. the Banach–Tarski one.

Some useful notions from algebraic topology

See also list of algebraic topology topics.

• Homology and cohomology: Betti numbers, Euler characteristic, degree of a continuous mapping.
• Operations: cup product, Massey product
• Intuitively attractive applications: Brouwer fixed-point theorem, Hairy ball theorem, Borsuk–Ulam theorem, Ham sandwich theorem.
• Homotopy groups (including the fundamental group).
• Chern classes, Stiefel–Whitney classes, Pontryagin classes.

Generalizations

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 certain structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.

See also

• List of algebraic topology topics
• List of general topology topics
• List of geometric topology topics
• List of topology topics
• Publications in topology
• Topology glossary
• General topology
• List of examples in general topology

References

Further reading

• Basener, William (2006). Topology and Its Applications (1st ed.). Wiley. ISBN 0-471-68755-3. 
• Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).
• Breitenberger, E. (2006). "Johann Benedict Listing". In I.M. James. History of Topology. North Holland. ISBN 978-0-444-82375-5. 
• Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6. 
• Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2. 
• Pickover, Clifford A. (2006). The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. Thunder's Mouth Press (Provides a popular introduction to topology and geometry). ISBN 1-56025-826-8. 
• Boto von Querenburg (2006). Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9 (German)
• Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6. 

External links

• Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov
• Topology at the Open Directory Project
• The Topological Zoo at The Geometry Center
• Topology Atlas
• Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas
• Topology Glossary
• Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.

UpToDate Contents

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• 1. インスリン受容体の構造および機能 structure and function of the insulin receptor

English Journal

• Choking loops on surfaces.

• Feng X, Tong Y.SourceMichigan State University, East Lansing.
• IEEE transactions on visualization and computer graphics.IEEE Trans Vis Comput Graph.2013 Aug;19(8):1298-306. doi: 10.1109/TVCG.2013.9.
• We present a method for computing "choking" loops-a set of surface loops that describe the narrowing of the volumes inside/outside of the surface and extend the notion of surface homology and homotopy loops. The intuition behind their definition is that a choking loop represents the region where an
• PMID 23744260

• Testing the use of ITS rDNA and protein-coding genes in the generic and species delimitation of the lichen genus Usnea (Parmeliaceae, Ascomycota).

• Truong C, Divakar PK, Yahr R, Crespo A, Clerc P.SourceConservatoire et Jardin botaniques de la Ville de Genève, CP 60, 1292 Chambésy, Switzerland; Laboratoire de systématique végétale et biodiversité, Faculté des sciences de l'Université de Genève, CP 60, 1292 Chambésy, Switzerland. Electronic address: camille.truong@ville-ge.ch.
• Molecular phylogenetics and evolution.Mol Phylogenet Evol.2013 Aug;68(2):357-72. doi: 10.1016/j.ympev.2013.04.005. Epub 2013 Apr 16.
• In lichen-forming fungi, traditional taxonomical concepts are frequently in conflict with molecular data, and identifying appropriate taxonomic characters to describe phylogenetic clades remains challenging in many groups. The selection of suitable markers for the reconstruction of solid phylogeneti
• PMID 23603312

• Species history and divergence times of viviparous and oviparous Chinese toad-headed sand lizards (Phrynocephalus) on the Qinghai-Tibetan Plateau.

• Jin YT, Brown RP.SourceDepartment of Biology, College of Life Sciences, China Jiliang University, Hangzhou 310018, Zhejiang, PR China.
• Molecular phylogenetics and evolution.Mol Phylogenet Evol.2013 Aug;68(2):259-68. doi: 10.1016/j.ympev.2013.03.022. Epub 2013 Apr 6.
• The Qinghai-Tibetan Plateau (QTP) is an important biogeographical area and has recently become a focus for biodiversity studies. Phyrnocephalus lizards form a widespread Eurasian group with oviparous and viviparous reproductive modes, but two previous mtDNA studies of species from around the QTP hav
• PMID 23567019

• Species tree phylogeny and character evolution in the genus Centipeda (Asteraceae): Evidence from DNA sequences from coding and non-coding loci from the plastid and nuclear genomes.

• Nylinder S, Cronholm B, de Lange PJ, Walsh N, Anderberg AA.SourceDepartment of Phanerogamic Botany, Swedish Museum of Natural History, P.O. Box 50007, SE-104 05 Stockholm, Sweden.
• Molecular phylogenetics and evolution.Mol Phylogenet Evol.2013 Aug;68(2):239-50. doi: 10.1016/j.ympev.2013.03.020. Epub 2013 Apr 2.
• A species tree phylogeny of the Australian/New Zealand genus Centipeda (Asteraceae) is estimated based on nucleotide sequence data. We analysed sequences of nuclear ribosomal DNA (ETS, ITS) and three plasmid loci (ndhF, psbA-trnH, and trnL-F) using the multi-species coalescent module in BEAST. A tot
• PMID 23558159

Japanese Journal

• Mobility of Ions Trapped Below a Free Surface of Superfluid

• Ikegami Hiroki,Chung Suk Bum,Kono Kimitoshi
• J Phys Soc Jpn 82(12), 124607-124607-12, 2013-12-15
• … Mobility measurements are conducted for two purposes: one is to figure out details of the scattering of a quasiparticle by an impurity in p-wave superfluids, and the other is to detect the surface Andreev bound states (SABSs) at a free surface of ³He-B formed as a result of the nontrivial topology of ³He-B. …
• NAID 150000107910

• 秘密分散法を用いたセキュアなWSNの構築

• 五百川 貴史,王 家宏,児玉 英一郎,高田 豊雄
• 情報処理学会研究報告. MBL, [モバイルコンピューティングとユビキタス通信研究会研究報告] 2013-MBL-69(14), 1-6, 2013-12-12
• 無線センサネットワーク (Wireless Sensor Networks: WSN) は重要なネットワーク技術の 1 つであり,この WSN は防災や防犯,施設の監視など様々な分野に利用されている.WSN の利用されるシステムにおいては,様々なレベルのセキュリティが求められている.そして,それは,ネットワークトポロジや WSN において使用されるセンサノードの品質などによって制約を受けている.本 …
• NAID 110009647113

• Discharge characteristics of multicell lithium-ion battery with nonuniform cells

• Miyatake So,Susuki Yoshihiko,Hikihara Takashi,Itoh Syuichi,Tanaka Kenichi
• Journal of Power Sources 241, 736-743, 2013-11
• … By using a multicell battery with two different types of cell, we experimentally investigate the relationship between discharge capacity and battery topology. …
• NAID 120005327031

• Controlling the stoichiometry and strand polarity of a tetramolecular G-quadruplex structure by using a DNA origami frame.

• Rajendran Arivazhagan,Endo Masayuki,Hidaka Kumi,Lan Thao Tran Phong,Mergny Jean-Louis,Sugiyama Hiroshi
• Nucleic acids research 41(18), 8738-8747, 2013-10
• … The conformational changes were carried out by incorporating two duplex DNAs, with G-G mismatch repeats in the middle, inside a DNA origami frame and monitoring the topology change of the strands. … However, our method using DNA origami could successfully control the strand orientations, topology and stoichiometry of the G-quadruplex. …
• NAID 120005347132

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Related Pictures

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★リンクテーブル★

リンク元	「form」「topological」「形態」「トポロジー」「位相幾何学」

「form」

　　[★]

• n.

• 形、型、形態、フォーム、形式、(分類学)品種、

Progression rates are variable and an acute aggressive form with death in 6 months can occur.(疾患の進展速度は症例により様々で、6ヶ月で死亡する急性増悪も起こりうる)(CASES.244)

• v.

• 形づくる、形成する、編成する、成立する

関

appearance、breed、configure、cultivar、figure、formation、mode、morphology、orchestrate、pattern、race、shape、topology、type、typed

　 　 　 　 　 　 　 　 　 　 　 　 　 　

「topological」

　　[★]

• adj.

• 形態的な、形態上の、位相幾何学の、トポロジーの、トポロジカルな

関

morphologic、morphological、morphologically、topologic、topologically、topology

　

「形態」

　　[★]

英

morphology、topology、form

関

位相幾何学、型、形、形式、形成、形態学、トポロジー、品種、編成、フォーム、成立、形づくる

「トポロジー」

　　[★]

英

topology、topological

関

位相幾何学、形態、形態的、形態上、トポロジカル

「位相幾何学」

　　[★]

英

topology、topological

関

形態、形態的、トポロジー、形態上、トポロジカル