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In mathematics, topology (from the Greek τόπος, ''place'', and λόγος, ''study''), is the study of a collection of open sets, making a given set a topological space. It is an area of mathematics concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. Important topological properties include connectedness and compactness.〔http://dictionary.reference.com/browse/topology〕 Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.〔http://www.math.wayne.edu/~rrb/topology.html〕 Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the ''geometria situs'' (Greek-Latin for "geometry of place") and ''analysis situs'' (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term ''topology'' was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology has many subfields: *General topology establishes the foundational aspects of topology and investigates properties of topological spaces and investigates concepts inherent to topological spaces. It includes point-set topology, which is the foundational topology used in all other branches (including topics like compactness and connectedness). *Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups. *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. *Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots. ==History== Topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on the Seven Bridges of Königsberg〔Euler, Leonhard, (Solutio problematis ad geometriam situs pertinentis )〕 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'',〔Listing, Johann Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848〕 who had used the word for ten years in correspondence before its first appearance in print. The English form topology was first used in 1883 in Listing's obituary in the journal ''Nature''〔Tait, Peter Guthrie, "Johann Benedict Listing (obituary)", Nature *27 *, 1 February 1883, pp. 316–317〕 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''. 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. 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. Henri Poincaré published ''Analysis Situs'' in 1895,〔Poincaré, Henri, "Analysis situs", Journal de l'École Polytechnique ser 2, 1 (1895) pp. 1–123〕 introducing the concepts of 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.〔Fréchet, Maurice, "Sur quelques points du calcul fonctionnel", PhD dissertation, 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.〔Hausdorff, Felix, "Grundzüge der Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)〕 Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. For further developments, see point-set topology and algebraic topology. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Topology」の詳細全文を読む スポンサード リンク
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