topology

1. Topographic study of a given place, especially the history of a region as indicated by its topography.
2. Medicine. The anatomical structure of a specific area or part of the body.
3. Mathematics. The study of the properties of geometric figures or solids that are not changed by homeomorphisms, such as stretching or bending. Donuts and picture frames are topologically equivalent, for example.
4. Computer Science. The arrangement in which the nodes of a LAN are connected to each other.

For more information on topology, visit Britannica.com.

The branch of mathematics that studies the qualitative properties of spaces, as opposed to the more delicate and refined geometric or analytic properties. While there are earlier results that belong to the field, the beginning of the subject as a separate branch of mathematics dates to the work of H. Poincaré during 1895–1904. The ideas and results of topology have a central place in mathematics, with connections to almost all the other areas of the subject.

The difference between topological and geometric properties is illustrated by the example of a space with three separate pieces. The exact shapes of the pieces constitute a geometric property of the space, and the study of these shapes is in the domain of differential geometry, but the fact that the space has three separate pieces is a qualitative or topological property. As another example, if a round sphere is deformed to be pear-shaped (or even more irregularly shaped, like the surface of the Earth), then the geometric notions of distance, straight line, and angle are changed, but the topological properties of the surface are left unchanged. However, if a handle is added by cutting two holes in the sphere and connecting them by a curved pipe, then the topology of the surface is changed (see illustration).

Process of adding a handle to a 2-sphere. (a) Cutting of holes in sphere. (b) Connecting of holes by a curved pipe.

Four major areas of topology are algebraic topology, homotopy theory, general topology, and manifold theory. Algebraic topology, the first area of modern topology to be developed, is concerned with associating algebraic invariants to geometric spaces in order to measure higher-dimensional analogs of the number of pieces of a space or the number of handles of a surface. (An algebraic invariant of a space is an algebraic object associated to the space that remains unchanged if the space is replaced by a homeomorphic space. By an algebraic object is meant either an algebraic structure, such as a group, ring, or field, or an element of an algebraic structure.) Algebraic topology has tremendous influence on other branches of mathematics, both direct (application of the invariants of algebraic topology to problems from other areas of mathematics and physics) and indirect (application in other contexts of ideas arising from algebraic topology).

As algebraic topology developed, it became clear that if one function could be continuously deformed to another (that is, if they were homotopic), then these two functions behaved in the same way as far as the invariants of algebraic topology were concerned. This led naturally to the study of invariants that remain unchanged as the maps are deformed by homotopies, that is, homotopy invariants. This study, which is an offshoot of algebraic topology, is called homotopy theory. Some of the most interesting homotopy invariants are the higher homotopy groups. These proved extremely difficult to compute, even for spaces as simple as the sphere, and are the subject of much investigation.

Early in the development of topology it was realized that the foundations of the subject needed attention. General or point-set topology studies the relationships between the basic topological properties that spaces may possess.

Before abstract topological spaces were defined, there were numerous examples of spaces arising from geometric and analytic problems. The most important of these is a class of spaces known as manifolds. Both because of their ubiquitous appearance throughout mathematics and because they possess extraordinarily rich topological properties, manifolds became one of the central objects of study in topology. The basic theme in manifold theory is to find sufficient algebraic invariants to classify, that is, to list comprehensively, all manifolds, and to give methods for evaluating these invariants in geometric cases. See also Manifold (mathematics).

The study of those properties of a geometric model, such as connectivity, which are not dependent on position.

Topology refers primarily to the branch of mathematics that rigorously treats questions of neighborhoods, limits, and continuity. Psychoanalysts have applied it to the study of unconscious structures.

In what have been called his two "topographies" (the first dating from 1900 and the second from 1923), Freud resorted to schemas to represent the various parts of the psychic apparatus and their interrelations. These schemas implicitly posited an equivalence between psychic and Euclidean space.

Early on, Jacques Lacan noted that the limitations of such a naive topology had restricted Freudian theory, not only in the description of the psychic apparatus (a description that in the end required an appeal to the economic point of view), but also in the specificity of clinical structures. The hypothesis that the unconscious is structured like a language, that is, in two dimensions, led Lacan to the topology of surfaces. The concept of foreclosure, for example, which he constructed on the basis of this topology, confirmed the heuristic value of his approach.

In his seminar "Identification" (1961-1962), Lacan unveiled a collection of topological objects—such as the torus, the Möbius strip, and the cross-cap—that served pedagogical aims. But already he saw them as more than just models. With the Borromean knot, introduced in 1973, he took the position that these objects were a real presentation of the subject and not just a representation. Below are several of Lacan's topological objects.

1. The Cut and the Signifier

Far from being given a priori, every space is organized on the basis of cuts and can actually be considered as a cut in the space of a higher dimension. We are familiar with the subjective impact of this: The events of our lives only become history through the castration complex, which organizes our reality at the price of an imaginary cutting off of the penis. According to Freud, by introjecting a single trait of another, the subject identifies with the other (at the price of losing this person as a love object). In the single trait Lacan found the very structure of the signifier: A cut allows the lost object to fall away. He called this cut the "unary trait."

The linguist Ferdinand de Saussure insisted on the fundamentally negative, purely differential character of the signifier. Lacan formalized this property in the double loop, or "interior eight," in which the gap created by the cut is closed after a second trip around a fictional axis. The difference of the signifier from itself is indicated by the difference between the two trips around the loop (Figure 1).

2. The Möbius Strip and Interpretation

If a signifier represents the subject for another signifier, then the subject would be supported by a surface whose edge would be a signifying cut. Note that the plane—the usual screen for the subject's images, figures, and dreams, that is, plans—is a surface that does not meet these conditions. The double loop cannot be drawn on a plane without showing a cut. The same is true of a sphere, a simple representation of the universe.

The Möbius strip, on the other hand, can represent this cut and symbolize the subject of the unconscious. Since a Möbius strip only has one surface, it is possible to pass from one side to the other without crossing over any edge—an apt representation of the return of the repressed. The Möbius strip also has certain other peculiarities. A cut that runs one-third from the edge and parallel to the edge divides the strip into a two-sided strip linked to what remains of the original Möbius strip. But if this cut is made in the center, it does not divide the Möbius strip in two. Instead, the entire strip is transformed into a strip with two sides. This characteristic illustrates the equivalence between the Möbius strip (the subject) and the medial cut that transforms it, and also provides a model of how interpretation functions. Interpretation does not abolish the unconscious. On the contrary, it makes the unconscious real for the subject by its transformed appearance as another (an Other) surface (figure 2).

3. The Torus

Lacan made different uses of the torus. By drawing Venn diagrams, traditionally used to illustrate basic logical operations, on the surface of the torus, he demonstrated the extent to which our thinking depends upon the plane surface, and he also provided another possible basis for the logic of the unconscious (Figure 3).

By inscribing the same circles on the surface of the torus, Lacan revealed the logic of the unconscious discovered by Freud (Figure 4).

On the torus, only symmetrical difference is consistent. Thus we have a demonstration of how the signifier can be different from all other signifiers and also from itself.

Lacan also used the torus to represent the subject as the subject of demand. In this sense, the torus can be conceived as the surface created by the iteration of the trajectory of the subject's demand. This trajectory turns around two different empty spaces, one that is "internal," D, the lack created in the real by speech, and one that is "central," d, corresponding to the place of the elusive object of desire that the drive goes around before completing the loop (Figure 5).

For every torus, there is a complementary torus, and the empty spaces of the two are the inverse of each other. Lacan made this structure of complementary toruses the support of the neurotic illusion that makes the demand of the Other the object of subject's desire and, conversely, makes the desire of the Other the object of subject's demand. This structure also arises from the fact that on a torus, the signifying cut (the double loop) does not detach any fragment. Neurotic subjects, insofar as they give in to neurosis, insofar as they are "in the torus," are not organized around their own castration, but instead excuse themselves by substituting the Other's demand for the object of their fantasy (figure 6).

4. The Cross-Cap

The cross-cap, or more precisely, the projective plane, can represent the subject of desire in relation to the lost object. A double loop drawn on its surface in effect divides this single-sided surface into two heterogeneous parts: a Möbius strip representing the subject and a disk representing object a, the cause of desire. The disk is centered on a point that is related to the irreducible singularity of this surface, which Lacan identified with the phallus. Unlike the representation of the subject produced on the torus, here a single cut, which symbolizes castration, produces both the subject and the object in its divisions (figure 7).

5. The Borromean Knot

Introduced by Lacan in 1973, the Borromean knot is the solution to a problem perceivable only in Lacanian theory but having extremely practical clinical applications. The problem is: How are the three registers posited as making up subjectivity—the real (R), the symbolic (S), and the imaginary (I)—held together?

Indeed, the symbolic (the signifier) and the imaginary (meaning) seem to have hardly anything in common—a fact demonstrated by the abundance and heterogeneity of languages. Moreover, the real, by definition, escapes the symbolic and the imaginary, since its resistance to them is precisely what makes it real.

This is why Lacan identified the real with the impossible.) In psychoanalysis, the real resists, and thus is distinct from, the imaginary defenses that the ego uses specifically to misrecognize the impossible and its consequences.

If each of the three registers R, S, and I that make up the Borromean knot is recognized to be toric in structure and the knot is constructed in three-dimensional space, it constitutes the perfect answer to the problem above, because it realizes a three-way joining of all three toruses, while none of them is actually linked to any other: If any one of them is cut, the other two are set free. Reciprocally, any knot that meets these conditions is called Borromean. Note that the subject is now defined by such a knot and not merely, as with the cross-cap, as the effect of a cut (figure 8).

Unfortunately, this ideal solution, which could be considered normal (without symptoms), seems to lead to paranoia. Lacan considered this to be the result of failure to distinguish among the three registers, as if they were continuous, which indeed occurs in clinical work. Being identical, R, S, and I are only differentiated by means of a "complication," a fourth ring that Lacan called the "sinthome." By making a ring with the three others, the sinthome (symptom) differentiates the three others by assuring their knotting (figure 9).

In this arrangement, the sinthome has the function of determining one of the rings. If it is attached to the symbolic, it plays the role of the paternal metaphor and its corollary, a neurotic symptom.

Lacan also drew upon non-Borromean knots, generated by "slips," or mistakes, in tying the knots. These allowed him to represent the status of subjects who are unattached to the imaginary or the real and who compensate for this with supplements (Lacan, 2001). In such cases the sinthome is maintained.

By using knots, Lacan was able to reveal his ongoing research without hiding its uncertainties. The value of the knots, which resist imaginary representation, is that they advance research that is not mere speculation and that they can grasp—at the cost of abandoning a grand synthesis—a few "bits of the real" (Lacan, 1976-1977, session of March 16, 1976). Even though he knew something about topology as practiced by mathematicians, Lacan advised his students "to use it stupidly" (Lacan, 1974-1975, session of December 17, 1974) as a remedy for our imaginary simplemindedness. He also recommended manually working with the knots by cutting surfaces and tying knots. Finally, for Lacan, topology had not only heuristic value but also valuable implications for psychoanalytic practice.

Bibliography

Bourbaki, Nicolas. (1994). Elements of the history of mathematics (John Meldrum, Trans.). Berlin: Springer-Verlag.

Darmon, Marc. (1990). Essais sur la topologie Lacanienne. Paris:Éditions de l'Association Freudienne Internationale.

Lacan, Jacques. (1975). La troisième, intervention de J. Lacan, le 31 octobre 1974. Lettres de l 'École Freudienne, 16, 178-203.

——. (1974-1975). Le séminaire, livre XXII, R.S.I. Ornicar? 2-5.

——. (1976-1977). Le séminaire XXIII, 1975-76: Le sinthome. Ornicar? 6-11.

——. (2001). Joyce: Le symptôme. In his Autres écrits. Paris: Seuil.

Pont, Jean-Claude. (1974). La topologie algébrique des origines à Poincaré. Paris: Presses Universitaires de France.

—BERNARD VANDERMERSCH

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 spatial properties that are preserved under continuous deformations of objects, for example, 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, and 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”), and which later acquired the modern name of topology. By the middle of the 20^th 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. For instance, the function y = x³ is a homeomorphism of the real line.

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.

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.

Contents 1 History 2 Elementary introduction 3 Mathematical definition 4 Topology topics 4.1 Some theorems in general topology 4.2 Some useful notions from algebraic topology 4.3 Generalizations 5 Topology in art and literature 6 See also 7 References 8 External links

History

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

Topology began with the investigation of certain questions in geometry. Euler's 1736 paper on 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, Vandenhoeck und Ruprecht, Göttingen, pp. 67, 1848, who had used the word for ten years in correspondence before its first appearance in print. "Topology," its English form, was introduced in 1883 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 setting down 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.

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)."

The term topology is also used to refer to a structure imposed upon a set X, a structure which 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 (homeomorphism) of a coffee cup 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.

In order 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 in uppercase sans-serif font (Myriad); left - homeomorphism, right - 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 any set and let T be a family of subsets of X. Then T is a topology on X iff

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

If T is a topology on X, then the pair (X, T) is called a topological space, and the notation X_T is used to denote a set X endowed with the particular topology T.

The open sets in X are defined to be the members of T; note that in general not all subsets of X need be in T. A subset of X is said to be closed if its complement is in T (i.e., its complement is open). A subset of X may be open, closed, both, or neither.

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 to be 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 m-manifold can be embedded in some Euclidean space Rⁿ.
• The continuous image of a connected space is connected.
• A metric space is Hausdorff, also normal and paracompact.
• 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, Furstenberg's proof of the infinitude of primes.

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 which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.

Topology in art and literature

• Some M. C. Escher works illustrate topological concepts, such as Möbius strips and non-orientable spaces.

See also

Topology portal

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

References

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (November 2009)

• 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 978-0444823755. 
• Kelly, 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

Wikimedia Commons has media related to: Topology

Wikibooks has more on the topic of Topology

• 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.

v • d • e Topics in Topology Fields Topological spaces • General (point set) topology • Set-theoretic topology • Algebraic topology • Homology theory • Cohomology theory • Differential topology • Geometric topology • Combinatorial topology • Continuum theory Key concepts Open set, closed set • Continuity • Compact space • Uniform spaces • Metric spaces • Hausdorff space • Homotopy theory • Homotopy groups • Fundamental group • Simplicial complexes • CW complexes • Exact sequence • Homological algebra • K-theory Lists and glossaries Glossary of general topology • List of topology topics • List of general topology topics • List of geometric topology topics • List of algebraic topology topics • Publications in topology

v • d • e Major fields of mathematics Logic · Set theory · Category theory · Algebra (elementary – linear – abstract) · Number theory · Analysis/Calculus · Geometry · Topology · Dynamical systems · Combinatorics · Game theory · Information theory · Numerical analysis · Optimization · Computation · Probability · Statistics · Trigonometry

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(мат.) топология

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地志学, 局部解剖学, 拓扑数学

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n. - 地誌學, 局部解剖學, 拓撲數學

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‏(الاسم) ألدراسه ألطوبوغرافيه لمكان معين, ألطوبولوجيا‏

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