4114:
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1902:
1121:, 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. However, the sphere is not homeomorphic to the doughnut.
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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
1206:
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 is
1256:, 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
292:
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
1268:
can all be made precise by using open sets. Several topologies can be defined on 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.
1397:
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 the space and affecting the curvature or volume.
251:) 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
855:
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, the
1351:
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 a convenient proof that any subgroup of a
330:
Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by
1790:
In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order which surrounds each piece and traverses each edge only once. This process is an application of the
1734:
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
1477:
is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
808:
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
405:, written in his native German, in 1847, having used 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
1535:, 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
281:
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
793:
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,
1416:(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
304:. 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.
3103:
378:
respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology.
270:." 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
1389:
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
1655:
topologies 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
1916:
339:
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
1550:
uses techniques from algebraic topology to determine the large scale structure of a set (for instance, determining if a cloud of points is spherical or
1597:
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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
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to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in
3574:
138:
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Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular,
832:"for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".
2793:
220:
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.
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are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are
1951:
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in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive
17:
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1458:(now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.
278:, the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes.
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1931:
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Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things,
1275:
are an important class of topological spaces where the distance between any two points is defined by a function called a
1063:
A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and of a (holeless) cow into a sphere
2198:
Adams, Colin Conrad, and Robert David
Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
1531:
classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings.
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The Möbius Strip: Dr. August Möbius's
Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology
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Aleksandrov, P.S. (1969) . "Chapter XVIII Topology". In
Aleksandrov, A.D.; Kolmogorov, A.N.; Lavrent'ev, M.A. (eds.).
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to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated".
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120:, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
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1820:
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325:
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1936:
57:
2982:(Provides a well motivated, geometric account of general topology, and shows the use of groupoids in discussing
2628:"Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry"
4130:
1581:
1295:. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the
1179:, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the
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252:
205:
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65:
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Listing, Johann
Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848
1767:, one finds paths between two points in configuration space. These paths represent a motion of the robot's
1429:
4057:
3613:
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1601:
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have been extensively applied to classify and compare the topology of folded proteins and nucleic acids.
1139:
is a topological space that resembles
Euclidean space near each point. More precisely, each point of an
162:
3628:
3429:
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1144:
1023:
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2146:
Hausdorff, Felix, "Grundzüge der
Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)
1966:
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1385:
More specifically, differential topology considers the properties and structures that require only a
185:
42:
31:
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1621:
1240:
801:. Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by
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to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.
266:
of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a
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point of view) and both separate the plane into two parts, the part inside and the part outside.
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Occasionally, one needs to use the tools of topology but a "set of points" is not available. In
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The topological dependence of mechanical properties in solids is of interest in disciplines of
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137:, which allows defining continuous deformation of subspaces, and, more generally, all kinds of
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Analyse these topological complexes via algebraic topology – specifically, via the theory of
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382:
170:
285:. The impossibility of crossing each bridge just once applies to any arrangement of bridges
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1971:
1921:
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1643:. Electrical and mechanical properties depend on the arrangement and network structures of
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802:
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158:
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of surface structures is the subject of interest with applications in multi-body physics.
8:
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3823:
3757:
3714:
3704:
3684:
3286:
3121:
2469:"Topological properties of a self-assembled electrical network via ab initio calculation"
2005:
Differential
Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems
1827:
1736:
1572:
Encode the persistent homology of a data set in the form of a parameterized version of a
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1084:
117:
106:
102:
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1279:. In a metric space, an open set is a union of open disks, where an open disk of radius
247:
demonstrated that it was impossible to find a route through the town of Königsberg (now
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Connect the shapes crochet motifs: creative techniques for joining motifs of all shapes
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1187:, which cannot (that is, all their realizations are surfaces that are not manifolds).
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2009:
1976:
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Geometric topology is a branch of topology that primarily focuses on low-dimensional
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846:
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coined the term "topological space" and gave the definition for what is now called a
235:, which have only one surface and one edge, are a kind of object studied in topology.
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124:
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on those categories, and with that the definition of general cohomology theories.
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to study topological spaces. The basic goal is to find algebraic invariants that
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829:
810:
798:
794:
166:
142:
3494:
2737:
Horak, Mathew (2006). "Disentangling
Topological Puzzles by Using Knot Theory".
2424:. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press.
232:
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Proceedings of the
National Academy of Sciences of the United States of America
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1740:
1739:, and then generalized in other areas of physics, for instance in photonics by
1440:
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880:
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113:
80:
1670:(or topological field theory or TQFT) is a quantum field theory that computes
4170:
3853:
3785:
3737:
3411:
3321:
3301:
3093:
2669:
2606:
The Shape of Space: How to
Visualize Surfaces and Three-dimensional Manifolds
2358:
The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
2327:
1792:
1717:
1698:
1694:
1686:
1682:
1609:
1585:
1454:/spherical, zero curvature/flat, and negative curvature/hyperbolic – and the
1300:
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1088:
861:
782:
770:
453:
297:
282:
178:
174:
150:
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2189:
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
1382:
and together they make up the geometric theory of differentiable manifolds.
1167:, are one-dimensional manifolds. Two-dimensional manifolds are also called
3795:
3790:
3694:
3396:
3316:
3262:
3108:
2916:
Breitenberger, E. (2006). "Johann Benedict Listing". In James, I.M. (ed.).
2677:
2579:
2510:
2133:
1706:
1573:
1304:
1272:
1180:
1148:
790:
766:
632:
458:
398:
286:
271:
256:
146:
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181:, which allows distinguishing a circle from two non-intersecting circles.
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2771:
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1678:
1532:
1524:
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248:
98:
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1782:
are based on topological aspects of the puzzle's shapes and components.
1337:
homeomorphism, though usually most classify up to homotopy equivalence.
3982:
3861:
3656:
3350:
3281:
3240:
1956:
1819:
focused on geometry and topology, and their applications, published by
1353:
1345:
1164:
1007:
865:
825:
344:
61:
38:
2894:, Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989,
2492:
1045:
3375:
2750:
2109:
2084:
2008:. Texts in Applied Mathematics. Vol. 18. Springer. p. 204.
1836:
which publishes papers of high quality and significance in topology,
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1644:
1451:
1296:
1235:
973:
857:
661:
312:
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1446:
Low-dimensional topology is strongly geometric, as reflected in the
293:
progressively enlarging it, while shrinking the hole into a handle.
228:
3805:
3732:
3360:
3328:
3277:
1837:
1756:
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surfaces are complex curves) – by the uniformization theorem every
1413:
1325:
Algebraic topology is a branch of mathematics that uses tools from
1130:
1110:
425:
154:
2542:
161:. The following are basic examples of topological properties: the
3671:
1882:. Undergraduate texts in mathematics. New York: Springer-Verlag.
1720:, as different manifolds can sustain different kinds of strings.
1326:
267:
3127:
3117:
1868:. Dover books on mathematics. Mineola, N.Y: Dover publications.
1069:
3112:
1219:
1160:
1073:
A continuous transformation can turn a coffee mug into a donut.
1054:
491:
414:
Their work was corrected, consolidated and greatly extended by
51:
1393:
that exist in differential topology. For instance, volume and
2873:
Euler's Gem: The Polyhedron Formula and the Birth of Topology
2526:"Three-dimensional structure of a sheet crumpled into a ball"
1768:
1752:
1551:
1334:
868:
can be thought of as the same set with different topologies.
516:
1554:). The main method used by topological data analysis is to:
157:. A property that is invariant under such deformations is a
3153:
2334:. Cambridge, MA: Harvard University, Dept. of Mathematics.
1723:
In cosmology, topology can be used to describe the overall
177:, which allows distinguishing between a line and a circle;
3038:(Provides a popular introduction to topology and geometry)
1033:
1014:
itself are always both closed and open. An open subset of
1917:
Characterizations of the category of topological spaces
1097:
or map from one topological space to another is called
2781:
A Topological Puzzle, Inta Bertuccioni, December 2003.
1659:
where the dependence of stiffness and friction on the
1494:
of open sets as the basic notion of the theory, while
3135:
Aisling McCluskey and Brian McMaster, Topology Atlas.
2800:
The Figure Eight Puzzle, Science and Math, June 2012.
418:. In 1895, he published his ground-breaking paper on
2220:(2002) Cambridge University Press, xii+544 pp.
1897:
84:
3019:
2791:https://www.futilitycloset.com/the-figure-8-puzzle
1854:(2nd ed.). Upper Saddle River, NJ: Prentice Hall.
1588:, are formalized using topology. In this context,
1428:, crumpling and the planar and higher-dimensional
1105:of any open set is open. If the function maps the
440:Topological characteristics of closed 2-manifolds
255:problem led to the branch of mathematics known as
212:are arguably the field's first theorems. The term
2524:Cambou, Anne Dominique; Narayanan, Menon (2011).
1620:Topology is relevant to physics in areas such as
4168:
2626:Haldane, F. D. M.; Raghu, S. (10 January 2008).
1612:(equivalently, finitely observable) properties.
1370:Differential topology is the field dealing with
1238:. The fundamental concepts of topology, such as
2700:Introduction to Robotics: Mechanics and Control
2523:
2383:
2379:
2377:
1002:(that is, its complement is open). A subset of
296:Homeomorphism can be considered the most basic
2841:Mathematics / Its Content, Methods and Meaning
2171:. The Norwegian Academy of Science and Letters
1709:for work related to topological field theory.
1558:Replace a set of data points with a family of
921:Any intersection of finitely many elements of
3568:
3169:
2915:
2466:
2393:Bulletin of the American Mathematical Society
2300:Bulletin of the American Mathematical Society
1727:. This area of research is commonly known as
998:is said to be closed if its complement is in
424:, which introduced the concepts now known as
401:. Listing introduced the term "Topologie" in
3144:Moscow 1935: Topology moving towards America
2625:
2374:
1685:in algebraic topology, and to the theory of
953:is called a topological space. The notation
3582:
2838:
2047:
2002:Hubbard, John H.; West, Beverly H. (1995).
2001:
1608:over open sets, which are characterized as
1340:The most important of these invariants are
1287:is the set of all points whose distance to
195:
189:
92:
3575:
3561:
3537:
3510:
3176:
3162:
2060:
2058:
2056:
1311:. Having a metric simplifies many proofs.
3041:
2910:Elements of Mathematics: General Topology
2643:
2569:
2559:
2541:
2500:
2404:
2311:
2290:
2161:
2126:Sur quelques points du calcul fonctionnel
2108:
1798:
1461:2-dimensional topology can be studied as
188:, who in the 17th century envisioned the
184:The ideas underlying topology go back to
3102:) is being considered for deletion. See
3046:(2nd ed.). Dover Publications Inc.
3014:
2866:
2043:
2027:
2025:
1359:
1068:
765:Unifying the work on function spaces of
311:
243:In one of the first papers in topology,
227:
165:, which allows distinguishing between a
49:
2419:
2123:
2083:Tait, Peter Guthrie (1 February 1883).
2053:
1771:and other parts into the desired pose.
1647:and elementary units in materials. The
1034:Continuous functions and homeomorphisms
14:
4169:
2934:
2809:
2711:
1600:, characterizes topological spaces as
3556:
3157:
2960:
2847:
2736:
2352:
2326:
2155:
2031:
2022:
1401:
1314:
1175:are manifolds. Examples include the
972:. By definition, every topology is a
968:endowed with the particular topology
840:
4137:
3088:Viro, Ivanov, Netsvetaev, Kharlamov.
2085:"Johann Benedict Listing (obituary)"
2082:
1932:List of examples in general topology
1840:, and adjacent areas of mathematics.
1774:
1498:are structures defined on arbitrary
1151:to the Euclidean space of dimension
97:, 'study') is the branch of
4149:
3086:Elementary Topology: A First Course
2241:
1562:, indexed by a proximity parameter.
1542:
1195:
432:, which are now considered part of
381:Further contributions were made by
131:endowed with a structure, called a
101:concerned with the properties of a
24:
2881:
2714:Invitation to Topological Robotics
2448:. Nobel Foundation. 4 October 2016
1712:The topological classification of
1481:
1075:Ceramic model by Keenan Crane and
25:
4193:
3106:to help reach a consensus. ›
3061:
2843:(2nd ed.). The M.I.T. Press.
2716:. European Mathematical Society.
2467:Stephenson, C.; et., al. (2017).
2446:"The Nobel Prize in Physics 2016"
2360:. American Mathematical Society.
2296:"The point of pointless topology"
1942:List of geometric topology topics
1927:List of algebraic topology topics
1803:
1214:, which are sets equipped with a
1010:), or neither. The empty set and
91:, 'place, location', and
4148:
4136:
4125:
4124:
4112:
3536:
3509:
3499:
3489:
3478:
3468:
3467:
3261:
2244:Introduction to Smooth Manifolds
1900:
1821:Mathematical Sciences Publishers
1668:topological quantum field theory
1053:
1044:
817:. For further developments, see
326:History of the separation axioms
4033:Computational complexity theory
3042:Gemignani, Michael C. (1990) .
2852:. Saunders College Publishing.
2832:
2803:
2784:
2765:
2730:
2705:
2692:
2619:
2599:
2586:
2517:
2460:
2438:
2413:
2346:
2320:
2313:10.1090/s0273-0979-1983-15080-2
2284:
2260:
2235:
2201:
2192:
2183:
1937:List of general topology topics
1509:
56:A three-dimensional model of a
50:
2662:10.1103/PhysRevLett.100.013904
2384:Gunnar Carlsson (April 2009).
2272:Handbook of Geometric Topology
2149:
2140:
2117:
2076:
2067:
2037:
1995:
1844:
1716:has important implications in
1582:programming language semantics
1435:In high-dimensional topology,
1113:. If a continuous function is
320:was a problem solved by Euler.
13:
1:
3133:Topology Course Lecture Notes
3091:
2876:. Princeton University Press.
2702:, 3rd Ed. Prentice-Hall, 2004
2608:2nd ed (Marcel Dekker, 1985,
2406:10.1090/S0273-0979-09-01249-X
1983:
1502:that allow the definition of
1210:The basic object of study is
1006:may be open, closed, both (a
223:
60:. The figure-eight knot is a
3183:
3004:, Dover Publications, 2000,
2772:http://sma.epfl.ch/Notes.pdf
1988:
1785:
1751:The possible positions of a
1576:, which is called a barcode.
1143:-dimensional manifold has a
1124:
964:may be used to denote a set
7:
3074:Encyclopedia of Mathematics
1893:
1817:mathematic research journal
1746:
1439:are a basic invariant, and
1378:. It is closely related to
835:
337:Seven Bridges of Königsberg
318:Seven Bridges of Königsberg
274:on the sphere. As with the
253:Seven Bridges of Königsberg
206:Seven Bridges of Königsberg
10:
4198:
4083:Films about mathematicians
3430:Banach fixed-point theorem
1850:Munkres, James R. (2000).
1615:
1514:
1490:one considers instead the
1405:
1363:
1318:
1199:
1128:
1082:
844:
323:
307:
85:
36:
29:
4106:
4056:
4013:
3923:
3885:
3852:
3804:
3776:
3723:
3670:
3652:Philosophy of mathematics
3627:
3592:
3463:
3420:
3384:
3270:
3259:
3191:
3026:. Thunder's Mouth Press.
2592:Yau, S. & Nadis, S.;
2124:Fréchet, Maurice (1906).
1967:Topological Galois theory
1878:Armstrong, M. A. (1983).
1864:Willard, Stephen (2016).
1548:Topological data analysis
1456:geometrization conjecture
1234:and (finite or infinite)
1190:
910:Any union of elements of
462:
457:
452:
447:
444:
186:Gottfried Wilhelm Leibniz
105:that are preserved under
93:
66:Alexander–Briggs notation
32:Topology (disambiguation)
4088:Recreational mathematics
3146:, a historical essay by
3104:templates for discussion
2912:, Addison–Wesley (1966).
2712:Farber, Michael (2008).
2594:The Shape of Inner Space
1952:Publications in topology
1622:condensed matter physics
1376:differentiable manifolds
1372:differentiable functions
891:is called a topology on
821:and algebraic topology.
813:as part of his study of
403:Vorstudien zur Topologie
335:. His 1736 paper on the
37:Not to be confused with
4182:Mathematical structures
3973:Mathematical statistics
3963:Mathematical psychology
3933:Engineering mathematics
3867:Algebraic number theory
2848:Croom, Fred H. (1989).
2777:1 November 2022 at the
2632:Physical Review Letters
2561:10.1073/pnas.1019192108
2420:Vickers, Steve (1996).
2332:Grothendieck topologies
2216:6 February 2012 at the
1947:List of topology topics
1811:Geometry & Topology
1780:Disentanglement puzzles
1689:in algebraic geometry.
1496:Grothendieck topologies
1356:is again a free group.
1218:, that is, a family of
1018:which contains a point
899:Both the empty set and
391:Johann Benedict Listing
298:topological equivalence
218:Johann Benedict Listing
145:, and, more generally,
4119:Mathematics portal
3968:Mathematical sociology
3948:Mathematical economics
3943:Mathematical chemistry
3872:Analytic number theory
3753:Differential equations
3485:Mathematics portal
3385:Metrics and properties
3371:Second-countable space
2967:Topology and Groupoids
2850:Principles of Topology
1799:Resources and research
1755:can be described by a
1672:topological invariants
1637:mechanical engineering
1592:, building on work by
1448:uniformization theorem
1437:characteristic classes
1080:
321:
236:
196:
190:
73:
18:Topology (Mathematics)
4098:Mathematics education
4028:Theory of computation
3748:Hypercomplex analysis
3016:Pickover, Clifford A.
2814:. Storey Publishing.
2810:Eckman, Edie (2012).
2242:Lee, John M. (2006).
1725:shape of the universe
1422:handle decompositions
1380:differential geometry
1366:Differential topology
1360:Differential topology
1185:real projective plane
1072:
383:Augustin-Louis Cauchy
315:
276:Bridges of Königsberg
231:
55:
27:Branch of mathematics
4078:Informal mathematics
3958:Mathematical physics
3953:Mathematical finance
3938:Mathematical biology
3877:Diophantine geometry
3440:Invariance of domain
3392:Euler characteristic
3366:Bundle (mathematics)
2984:van Kampen's theorem
2739:Mathematics Magazine
2596:, Basic Books, 2010.
1972:Topological geometry
1922:Equivariant topology
1714:Calabi–Yau manifolds
1649:compressive strength
1626:quantum field theory
1580:Several branches of
1560:simplicial complexes
1395:Riemannian curvature
1207:point-set topology.
803:Kazimierz Kuratowski
302:homotopy equivalence
272:tangent vector field
159:topological property
30:For other uses, see
4093:Mathematics and art
4003:Operations research
3758:Functional analysis
3450:Tychonoff's theorem
3445:Poincaré conjecture
3199:General (point-set)
3122:The Geometry Center
3118:The Topological Zoo
3069:"Topology, general"
3044:Elementary Topology
2918:History of Topology
2796:25 May 2017 at the
2654:2008PhRvL.100a3904H
2552:2011PNAS..10814741C
2536:(36): 14741–14745.
2485:2017NatSR...741621S
2386:"Topology and data"
2292:Johnstone, Peter T.
2246:. Springer-Verlag.
2210:Algebraic topology.
2169:"Prize winner 2022"
2101:1883Natur..27..316P
1828:Journal of Topology
1761:configuration space
1737:quantum Hall effect
1567:persistent homology
1333:topological spaces
1303:, real and complex
1171:, although not all
1085:Continuous function
464:Torsion coefficient
441:
4038:Numerical analysis
3647:Mathematical logic
3642:Information theory
3435:De Rham cohomology
3356:Polyhedral complex
3346:Simplicial complex
2422:Topology via Logic
1908:Mathematics portal
1834:scientific journal
1729:spacetime topology
1630:physical cosmology
1488:pointless topology
1430:Schönflies theorem
1408:Geometric topology
1402:Geometric topology
1321:Algebraic topology
1315:Algebraic topology
1212:topological spaces
1081:
841:Topologies on sets
819:point-set topology
439:
434:algebraic topology
349:polyhedron formula
347:. This led to his
322:
264:hairy ball theorem
237:
216:was introduced by
210:polyhedron formula
74:
4164:
4163:
3763:Harmonic analysis
3550:
3549:
3339:fundamental group
3139:Topology Glossary
3053:978-0-486-66522-1
3033:978-1-56025-826-1
2998:Wacław Sierpiński
2977:978-1-4196-2722-4
2953:978-0-387-90125-1
2927:978-0-444-82375-5
2920:. North Holland.
2888:Ryszard Engelking
2859:978-0-03-029804-2
2493:10.1038/srep41621
2367:978-0-8218-3678-1
2274:, North-Holland.
2253:978-0-387-95448-6
2015:978-0-387-94377-0
1977:Topological order
1962:Topology glossary
1888:978-0-387-90839-7
1874:978-0-486-43479-7
1860:978-0-13-181629-9
1775:Games and puzzles
1763:. In the area of
1657:Contact mechanics
1641:materials science
1465:in one variable (
1443:is a key theory.
1262:arbitrarily small
937:is a topology on
925:is an element of
914:is an element of
875:be a set and let
847:Topological space
763:
762:
125:topological space
58:figure-eight knot
16:(Redirected from
4189:
4152:
4151:
4140:
4139:
4128:
4127:
4117:
4116:
4048:Computer algebra
4023:Computer science
3743:Complex analysis
3577:
3570:
3563:
3554:
3553:
3540:
3539:
3513:
3512:
3503:
3493:
3483:
3482:
3471:
3470:
3265:
3178:
3171:
3164:
3155:
3154:
3082:
3057:
3037:
3025:
3002:General Topology
2981:
2957:
2940:General Topology
2931:
2892:General Topology
2877:
2863:
2844:
2826:
2825:
2807:
2801:
2788:
2782:
2769:
2763:
2762:
2751:10.2307/27642974
2734:
2728:
2727:
2709:
2703:
2696:
2690:
2689:
2647:
2645:cond-mat/0503588
2623:
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2114:
2112:
2110:10.1038/027316a0
2095:(692): 316–317.
2080:
2074:
2071:
2065:
2062:
2051:
2048:Aleksandrov 1969
2041:
2035:
2029:
2020:
2019:
1999:
1910:
1905:
1904:
1866:General topology
1681:, the theory of
1606:Heyting algebras
1598:Michael B. Smyth
1543:Computer science
1529:Circuit topology
1521:circuit topology
1463:complex geometry
1387:smooth structure
1344:, homology, and
1309:Euclidean spaces
1294:
1290:
1286:
1282:
1202:General topology
1196:General topology
1154:
1142:
1101:if the inverse
1057:
1048:
1029:
1021:
1017:
1013:
1005:
1001:
997:
993:
985:
976:
971:
967:
963:
952:
941:, then the pair
940:
936:
928:
924:
917:
913:
906:
903:are elements of
902:
894:
890:
886:
878:
874:
779:Jacques Hadamard
753:
734:
720:
713:
707:
704:2-Manifold with
693:
679:
670:
660:
608:Projective plane
596:
582:
573:
566:
442:
438:
395:Bernhard Riemann
377:
373:
369:
365:
199:
193:
143:Euclidean spaces
103:geometric object
96:
95:
90:
89:
54:
21:
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4188:
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4167:
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4111:
4102:
4052:
4009:
3988:Systems science
3919:
3915:Homotopy theory
3881:
3848:
3800:
3772:
3719:
3666:
3637:Category theory
3623:
3588:
3581:
3551:
3546:
3477:
3459:
3455:Urysohn's lemma
3416:
3380:
3266:
3257:
3229:low-dimensional
3187:
3182:
3148:Hassler Whitney
3107:
3067:
3064:
3054:
3034:
2988:covering spaces
2978:
2954:
2944:Springer-Verlag
2936:Kelley, John L.
2928:
2884:
2882:Further reading
2860:
2835:
2830:
2829:
2822:
2808:
2804:
2798:Wayback Machine
2789:
2785:
2779:Wayback Machine
2770:
2766:
2735:
2731:
2724:
2710:
2706:
2698:John J. Craig,
2697:
2693:
2624:
2620:
2604:
2600:
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2587:
2522:
2518:
2465:
2461:
2451:
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2418:
2414:
2388:
2382:
2375:
2368:
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2325:
2321:
2289:
2285:
2266:R. B. Sher and
2265:
2261:
2254:
2240:
2236:
2218:Wayback Machine
2207:Allen Hatcher,
2206:
2202:
2197:
2193:
2188:
2184:
2174:
2172:
2167:
2166:
2162:
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2150:
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2141:
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2118:
2081:
2077:
2072:
2068:
2064:Richeson (2008)
2063:
2054:
2042:
2038:
2030:
2023:
2016:
2000:
1996:
1991:
1986:
1981:
1906:
1899:
1896:
1847:
1806:
1801:
1788:
1777:
1765:motion planning
1749:
1618:
1594:Samson Abramsky
1545:
1537:electrophoresis
1517:
1512:
1484:
1482:Generalizations
1471:conformal class
1410:
1404:
1368:
1362:
1342:homotopy groups
1323:
1317:
1292:
1288:
1284:
1280:
1204:
1198:
1193:
1152:
1140:
1133:
1127:
1091:
1083:Main articles:
1074:
1067:
1066:
1065:
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1060:
1059:
1058:
1050:
1049:
1036:
1027:
1019:
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1011:
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995:
991:
983:
982:The members of
974:
969:
965:
962:
954:
942:
938:
934:
926:
922:
915:
911:
904:
900:
892:
888:
884:
876:
872:
849:
843:
838:
830:Dennis Sullivan
828:was awarded to
811:Euclidean space
799:Hausdorff space
795:Felix Hausdorff
789:introduced the
787:Maurice Fréchet
743:
724:
715:
711:
709:
705:
688:
674:
665:
658:
591:
577:
571:
564:
486:
480:
474:
387:Ludwig Schläfli
375:
371:
367:
352:
328:
310:
262:Similarly, the
226:
191:geometria situs
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35:
28:
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11:
5:
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4095:
4090:
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4075:
4074:
4073:
4066:Mathematicians
4062:
4060:
4058:Related topics
4054:
4053:
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4030:
4025:
4019:
4017:
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3995:
3993:Control theory
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3771:
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3768:Measure theory
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3442:
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3418:
3417:
3415:
3414:
3409:
3404:
3402:Winding number
3399:
3394:
3388:
3386:
3382:
3381:
3379:
3378:
3373:
3368:
3363:
3358:
3353:
3348:
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3342:
3341:
3336:
3334:homotopy group
3326:
3325:
3324:
3319:
3314:
3309:
3304:
3294:
3289:
3284:
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3272:
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3201:
3195:
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3180:
3173:
3166:
3158:
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3151:
3141:
3136:
3130:
3128:Topology Atlas
3125:
3115:
3089:
3083:
3063:
3062:External links
3060:
3059:
3058:
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3032:
3012:
2995:
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2879:
2878:
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2858:
2845:
2834:
2831:
2828:
2827:
2821:978-1603429733
2820:
2802:
2783:
2764:
2745:(5): 368–375.
2729:
2723:978-3037190548
2722:
2704:
2691:
2618:
2598:
2585:
2516:
2459:
2437:
2431:978-0521576512
2430:
2412:
2399:(2): 255–308.
2395:. New Series.
2373:
2366:
2345:
2328:Artin, Michael
2319:
2283:
2268:R. J. Daverman
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2252:
2234:
2200:
2191:
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2046:, p. 63;
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1880:Basic topology
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1824:
1805:
1804:Major journals
1802:
1800:
1797:
1787:
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1773:
1748:
1745:
1683:four-manifolds
1661:dimensionality
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1544:
1541:
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1483:
1480:
1441:surgery theory
1426:local flatness
1406:Main article:
1403:
1400:
1364:Main article:
1361:
1358:
1319:Main article:
1316:
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1200:Main article:
1197:
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1129:Main article:
1126:
1123:
1077:Henry Segerman
1062:
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1040:
1039:
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994:. A subset of
958:
931:
930:
919:
908:
883:of subsets of
871:Formally, let
845:Main article:
842:
839:
837:
834:
815:Fourier series
761:
760:
757:
754:
741:
738:
737:Non-orientable
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682:Non-orientable
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640:Non-orientable
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615:Non-orientable
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567:-holed torus (
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451:
446:
421:Analysis Situs
416:Henri Poincaré
333:Leonhard Euler
309:
306:
245:Leonhard Euler
225:
222:
202:Leonhard Euler
197:analysis situs
151:homeomorphisms
69:
26:
9:
6:
4:
3:
2:
4194:
4183:
4180:
4178:
4175:
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4021:
4020:
4018:
4016:
4015:Computational
4012:
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3996:
3994:
3991:
3990:
3989:
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3878:
3875:
3873:
3870:
3868:
3865:
3863:
3860:
3859:
3857:
3855:
3854:Number theory
3851:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3822:
3820:
3817:
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3812:
3811:
3809:
3807:
3803:
3797:
3794:
3792:
3789:
3787:
3786:Combinatorics
3784:
3783:
3781:
3779:
3775:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
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3744:
3741:
3739:
3738:Real analysis
3736:
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3726:
3722:
3716:
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3436:
3433:
3431:
3428:
3427:
3425:
3423:
3419:
3413:
3412:Orientability
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3389:
3387:
3383:
3377:
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3327:
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3308:
3305:
3303:
3300:
3299:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3279:
3276:
3275:
3273:
3269:
3264:
3254:
3251:
3249:
3248:Set-theoretic
3246:
3242:
3239:
3238:
3237:
3234:
3230:
3227:
3226:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3209:Combinatorial
3207:
3205:
3202:
3200:
3197:
3196:
3194:
3190:
3186:
3179:
3174:
3172:
3167:
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3160:
3159:
3156:
3149:
3145:
3142:
3140:
3137:
3134:
3131:
3129:
3126:
3123:
3119:
3116:
3114:
3110:
3105:
3101:
3100:
3095:
3090:
3087:
3084:
3080:
3076:
3075:
3070:
3066:
3065:
3055:
3049:
3045:
3040:
3035:
3029:
3024:
3023:
3017:
3013:
3011:
3010:0-486-41148-6
3007:
3003:
2999:
2996:
2993:
2989:
2985:
2979:
2973:
2970:. Booksurge.
2969:
2968:
2963:
2962:Brown, Ronald
2959:
2955:
2949:
2945:
2941:
2937:
2933:
2929:
2923:
2919:
2914:
2911:
2907:
2904:
2901:
2900:3-88538-006-4
2897:
2893:
2889:
2886:
2885:
2875:
2874:
2869:
2865:
2861:
2855:
2851:
2846:
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2837:
2836:
2823:
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2806:
2799:
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2787:
2780:
2776:
2773:
2768:
2760:
2756:
2752:
2748:
2744:
2740:
2733:
2725:
2719:
2715:
2708:
2701:
2695:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2655:
2651:
2646:
2641:
2638:(1): 013904.
2637:
2633:
2629:
2622:
2615:
2614:0-8247-7437-X
2611:
2607:
2602:
2595:
2589:
2581:
2577:
2572:
2567:
2562:
2557:
2553:
2549:
2544:
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2535:
2531:
2527:
2520:
2512:
2508:
2503:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2463:
2447:
2441:
2433:
2427:
2423:
2416:
2407:
2402:
2398:
2394:
2387:
2380:
2378:
2369:
2363:
2359:
2355:
2349:
2341:
2337:
2333:
2329:
2323:
2314:
2309:
2305:
2301:
2297:
2293:
2287:
2281:
2280:0-444-82432-4
2277:
2273:
2269:
2263:
2255:
2249:
2245:
2238:
2231:
2230:0-521-79540-0
2227:
2226:0-521-79160-X
2223:
2219:
2215:
2212:
2211:
2204:
2195:
2186:
2170:
2164:
2158:, p. 129
2157:
2152:
2143:
2135:
2131:
2127:
2120:
2111:
2106:
2102:
2098:
2094:
2090:
2086:
2079:
2070:
2061:
2059:
2057:
2050:, p. 204
2049:
2045:
2044:Richeson 2008
2040:
2033:
2028:
2026:
2017:
2011:
2007:
2006:
1998:
1994:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1914:
1909:
1903:
1898:
1889:
1885:
1881:
1877:
1875:
1871:
1867:
1863:
1861:
1857:
1853:
1849:
1848:
1839:
1835:
1831:
1829:
1825:
1822:
1818:
1814:
1812:
1808:
1807:
1796:
1794:
1793:Eulerian path
1783:
1781:
1772:
1770:
1766:
1762:
1758:
1754:
1744:
1742:
1741:F.D.M Haldane
1738:
1732:
1730:
1726:
1721:
1719:
1718:string theory
1715:
1710:
1708:
1707:Fields Medals
1705:have all won
1704:
1700:
1696:
1692:
1688:
1687:moduli spaces
1684:
1680:
1675:
1673:
1669:
1664:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1633:
1631:
1627:
1623:
1613:
1611:
1610:semidecidable
1607:
1603:
1599:
1595:
1591:
1590:Steve Vickers
1587:
1586:domain theory
1583:
1575:
1571:
1568:
1564:
1561:
1557:
1556:
1555:
1553:
1549:
1540:
1538:
1534:
1530:
1526:
1522:
1507:
1505:
1501:
1497:
1493:
1489:
1479:
1476:
1472:
1468:
1464:
1459:
1457:
1453:
1449:
1444:
1442:
1438:
1433:
1431:
1427:
1423:
1419:
1418:orientability
1415:
1409:
1399:
1396:
1392:
1388:
1383:
1381:
1377:
1373:
1367:
1357:
1355:
1349:
1347:
1343:
1338:
1336:
1332:
1328:
1322:
1312:
1310:
1306:
1305:vector spaces
1302:
1301:complex plane
1298:
1291:is less than
1278:
1274:
1273:Metric spaces
1270:
1267:
1263:
1259:
1255:
1254:
1253:connectedness
1249:
1248:
1243:
1242:
1237:
1233:
1232:intersections
1230:under finite
1229:
1225:
1221:
1217:
1213:
1208:
1203:
1188:
1186:
1182:
1178:
1174:
1170:
1166:
1165:figure eights
1162:
1158:
1150:
1146:
1138:
1132:
1122:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1090:
1089:homeomorphism
1086:
1078:
1071:
1056:
1047:
1038:
1031:
1025:
1009:
989:
980:
978:
961:
957:
950:
946:
920:
909:
898:
897:
896:
882:
869:
867:
863:
862:complex plane
859:
854:
848:
833:
831:
827:
822:
820:
816:
812:
806:
804:
800:
796:
792:
788:
784:
783:Giulio Ascoli
780:
776:
775:Cesare Arzelà
772:
771:Vito Volterra
768:
758:
755:
751:
747:
742:
739:
736:
732:
728:
723:
718:
703:
702:
698:
695:
691:
687:
684:
681:
678:
673:
668:
663:
656:
655:
651:
648:
645:
642:
639:
636:
634:
631:
630:
626:
623:
620:
617:
614:
611:
609:
606:
605:
601:
598:
595:
590:
587:
584:
581:
576:
570:
563:
562:
558:
555:
552:
549:
546:
543:
541:2-holed torus
540:
539:
535:
532:
529:
526:
523:
520:
518:
515:
514:
510:
507:
504:
501:
498:
495:
493:
490:
489:
482:
476:
470:
469:
465:
460:
459:Betti numbers
455:
454:Orientability
450:
443:
437:
435:
431:
427:
423:
422:
417:
412:
410:
409:
404:
400:
396:
392:
388:
384:
379:
363:
359:
355:
350:
346:
342:
338:
334:
327:
319:
314:
305:
303:
300:. Another is
299:
294:
290:
288:
284:
283:homeomorphism
279:
277:
273:
269:
265:
260:
258:
254:
250:
246:
241:
234:
233:Möbius strips
230:
221:
219:
215:
211:
207:
203:
198:
192:
187:
182:
180:
179:connectedness
176:
172:
168:
164:
160:
156:
152:
148:
147:metric spaces
144:
140:
136:
135:
130:
126:
121:
119:
115:
111:
108:
104:
100:
88:
82:
78:
67:
63:
59:
53:
48:
44:
40:
33:
19:
4153:
4141:
4129:
4110:
4043:Optimization
3905:Differential
3886:
3829:Differential
3796:Order theory
3791:Graph theory
3695:Group theory
3542:Publications
3407:Chern number
3397:Betti number
3280: /
3271:Key concepts
3219:Differential
3184:
3097:
3072:
3043:
3021:
3001:
2992:orbit spaces
2966:
2939:
2917:
2909:
2891:
2872:
2868:Richeson, D.
2849:
2840:
2833:Bibliography
2811:
2805:
2786:
2767:
2742:
2738:
2732:
2713:
2707:
2699:
2694:
2635:
2631:
2621:
2605:
2601:
2593:
2588:
2533:
2529:
2519:
2476:
2472:
2462:
2450:. Retrieved
2440:
2421:
2415:
2396:
2392:
2357:
2354:Adams, Colin
2348:
2331:
2322:
2306:(1): 41–53.
2303:
2299:
2286:
2271:
2262:
2243:
2237:
2209:
2203:
2194:
2185:
2173:. Retrieved
2163:
2151:
2142:
2125:
2119:
2092:
2088:
2078:
2069:
2039:
2004:
1997:
1879:
1865:
1851:
1826:
1809:
1789:
1778:
1750:
1733:
1722:
1711:
1676:
1665:
1634:
1619:
1579:
1574:Betti number
1546:
1518:
1510:Applications
1485:
1460:
1445:
1434:
1411:
1391:deformations
1384:
1369:
1350:
1339:
1324:
1283:centered at
1276:
1271:
1265:
1261:
1257:
1251:
1245:
1239:
1223:
1209:
1205:
1181:Klein bottle
1149:homeomorphic
1145:neighborhood
1136:
1134:
1107:real numbers
1098:
1092:
1037:
1024:neighborhood
1022:is called a
987:
981:
959:
955:
948:
944:
932:
870:
852:
850:
823:
807:
791:metric space
785:and others,
767:Georg Cantor
764:
749:
745:
730:
726:
716:
714:cross-caps (
689:
676:
666:
657:Sphere with
633:Klein bottle
593:
579:
419:
413:
407:
402:
399:Enrico Betti
380:
361:
357:
353:
340:
329:
295:
291:
287:homeomorphic
280:
275:
261:
257:graph theory
242:
238:
213:
208:problem and
183:
132:
122:
110:deformations
76:
75:
47:
4155:WikiProject
3998:Game theory
3978:Probability
3715:Homological
3705:Multilinear
3685:Commutative
3662:Type theory
3629:Foundations
3585:mathematics
3505:Wikiversity
3422:Key results
3092:‹ The
2034:, p. 7
1845:Major books
1679:knot theory
1533:Knot theory
1525:knot theory
1247:compactness
1226:, which is
986:are called
249:Kaliningrad
175:compactness
99:mathematics
64:and has an
4171:Categories
3983:Statistics
3862:Arithmetic
3824:Arithmetic
3690:Elementary
3657:Set theory
3351:CW complex
3292:Continuity
3282:Closed set
3241:cohomology
2452:12 October
2340:0208.48701
2156:Croom 1989
2032:Croom 1989
1984:References
1957:Topoisomer
1703:Kontsevich
1584:, such as
1500:categories
1354:free group
1346:cohomology
1241:continuity
1163:, but not
1115:one-to-one
1099:continuous
1008:clopen set
866:Cantor set
864:, and the
826:Abel Prize
662:cross-caps
585:Orientable
547:Orientable
524:Orientable
499:Orientable
345:polyhedron
324:See also:
224:Motivation
155:homotopies
139:continuity
114:stretching
112:, such as
107:continuous
79:(from the
62:prime knot
39:topography
3910:Geometric
3900:Algebraic
3839:Euclidean
3814:Algebraic
3710:Universal
3530:geometric
3525:algebraic
3376:Cobordism
3312:Hausdorff
3307:connected
3224:Geometric
3214:Continuum
3204:Algebraic
3079:EMS Press
2670:0031-9007
2543:1203.5826
2479:: 41621.
1989:Citations
1786:Fiber art
1691:Donaldson
1645:molecules
1452:curvature
1414:manifolds
1297:real line
1266:far apart
1224:open sets
1222:, called
1125:Manifolds
988:open sets
858:real line
851:The term
824:The 2022
449:Euler num
163:dimension
4177:Topology
4131:Category
3887:Topology
3834:Discrete
3819:Analytic
3806:Geometry
3778:Discrete
3733:Calculus
3725:Analysis
3680:Abstract
3619:Glossary
3602:Timeline
3495:Wikibook
3473:Category
3361:Manifold
3329:Homotopy
3287:Interior
3278:Open set
3236:Homology
3185:Topology
3109:Topology
3094:template
3081:. 2001 .
3018:(2006).
2964:(2006).
2938:(1975).
2906:Bourbaki
2870:(2008).
2794:Archived
2775:Archived
2759:27642974
2686:44745453
2678:18232766
2580:21873249
2511:28155863
2473:Sci. Rep
2356:(2004).
2330:(1962).
2294:(1983).
2270:(2002),
2214:Archived
2175:23 March
1894:See also
1852:Topology
1838:geometry
1757:manifold
1747:Robotics
1653:crumpled
1552:toroidal
1331:classify
1216:topology
1173:surfaces
1169:surfaces
1147:that is
1137:manifold
1131:Manifold
1111:calculus
1095:function
853:topology
836:Concepts
466:(1-dim)
445:Manifold
430:homology
426:homotopy
214:topology
134:topology
118:twisting
77:Topology
43:typology
4143:Commons
3925:Applied
3895:General
3672:Algebra
3597:History
3520:general
3322:uniform
3302:compact
3253:Digital
3096:below (
2650:Bibcode
2571:3169141
2548:Bibcode
2502:5290745
2481:Bibcode
2134:8897542
2097:Bibcode
1759:called
1616:Physics
1602:Boolean
1515:Biology
1504:sheaves
1492:lattice
1475:metrics
1467:Riemann
1327:algebra
1220:subsets
1161:circles
977:-system
887:. Then
366:(where
308:History
268:cowlick
171:surface
3844:Finite
3700:Linear
3607:Future
3583:Major
3515:Topics
3317:metric
3192:Fields
3113:Curlie
3099:Curlie
3050:
3030:
3008:
2990:, and
2974:
2950:
2924:
2898:
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