# Topology

Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. Topology builds on set theory, considering both sets of points and families of sets.

The word topology is used both for the area of study and for a family of sets with certain properties described below that are used to define a topological space. Of particular importance in the study of topology are functions or maps that are homeomorphisms. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together.

When the discipline was first properly founded, toward the end of the 19th century, it was called geometria situs (Latin geometry of place) and analysis situs (Latin analysis of place). From around 1925 to 1975 it was an important growth area within mathematics.

Topology is a large branch of mathematics that includes many subfields. The most basic division within topology is point-set topology, which investigates such concepts as compactness, connectedness, and countability; algebraic topology, which investigates such concepts as homotopy, homology; and geometric topology, which studies manifolds and their embeddings, including knot theory.

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.

The branch of mathematics now called topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on Seven Bridges of Königsberg is regarded as one of the first topological results.

The term "topologie" was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie, Vandenhoeck und Ruprecht, Göttingen, pp. 67, 1848. However, Listing had already used the word for ten years in correspondence. "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.

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

A continuous deformation (homotopy) of a coffee cup into a doughnut (torus) and back.
Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics. General topology, or point-set topology, defines and studies properties of spaces and maps such as connectedness, compactness and continuity. Algebraic topology uses structures from abstract algebra, especially the group to study topological spaces and the maps between them.

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.

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair on a ball smooth." 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 blob (subject to certain conditions on the smoothness of the surface), 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 topological equivalence. The impossibility of crossing each bridge just once applies to any arrangement of bridges topologically equivalent to those in Königsberg, and the hairy ball theorem applies to any space topologically equivalent to a sphere.

Intuitively, two spaces are topologically equivalent if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist can't tell the coffee mug out of which he is drinking from the doughnut he is eating, 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 simple introductory exercise is to classify the lowercase letters of the English alphabet according to topological equivalence. (The lines of the letters are assumed to have non-zero width.) In most fonts in modern use, there is a class {a, b, d, e, o, p, q} of letters with one hole, a class {c, f, h, k, l, m, n, r, s, t, u, v, w, x, y, z} of letters without a hole, and a class {i, j} of letters consisting of two pieces. g may either belong in the class with one hole, or (in some fonts) it may be the sole element of a class of letters with two holes, depending on whether or not the tail is closed. For a more complicated exercise, it may be assumed that the lines have zero width; one can get several different classifications depending on which font is used. Letter topology is of practical relevance in stencil typography: The font Braggadocio, for instance, can be cut out of a plane without falling apart.

## 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 if
1. Both the empty set and X are elements of T.
2. Any union of 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 X together with T is called a topological space.

All sets in T are called open; note that not all subsets of X are in T. A subset of X is said to be closed if its complement is in T (i.e., it 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 space 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.

## Some theorems in general topology

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

## 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.

## References

• James Munkres (1999). Topology, 2nd edition, Prentice Hall. ISBN 0-13-181629-2.
• John L. Kelley (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
• Clifford A. Pickover (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.
• Querenburg, Boto von, (2006), Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9

Greek}}}
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
Set theory is the mathematical theory of sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary school, often as Venn diagrams, of collections of objects, and the elements of, and membership in, such collections.

Topological spaces are mathematical structures that allow the formalization of concepts such as convergence, connectedness and continuity.
function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
In the mathematical field of topology, a homeomorphism or topological isomorphism
The 19th Century (also written XIX century) lasted from 1801 through 1900 in the Gregorian calendar. It is often referred to as the "1800s.
Latin}}}
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Latin}}}
Official status
Official language of: Vatican City
Used for official purposes, but not spoken in everyday speech
Regulated by: Opus Fundatum Latinitas
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ISO 639-1: la
ISO 639-2: lat
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them. It is distinct from other branches of topology in that the topological spaces may be very general, and do not have
Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).
In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open spaces.
In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.
homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space (singular homology)
In mathematics, geometric topology is the study of manifolds and their embeddings. Low-dimensional topology, concerning questions of dimensions up to four, is a part of geometric topology.
Manifold may refer to:
• Manifold, an abstract mathematical space which, in a close-up view, resembles the spaces described by Euclidean geometry.
• Manifold (automotive engineering), an engine part which joins many connections into one, for example the exhaust manifold

Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3.
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Accumulation point: See limit point.

Alexandrov topology: A space X has the Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X

Topological spaces are mathematical structures that allow the formalization of concepts such as convergence, connectedness and continuity.
Leonhard Euler

Portrait by Johann Georg Brucker
Born March 15 1707
Basel, Switzerland
Died September 18 [O.S.
17th century - 18th century - 19th century
1700s  1710s  1720s  - 1730s -  1740s  1750s  1760s
1733 1734 1735 - 1736 - 1737 1738 1739

:
Subjects:     Archaeology - Architecture -
Seven Bridges of Königsberg is a famous solved mathematics problem inspired by an actual place and situation. The city of Königsberg, Prussia (now Kaliningrad, Russia) is set on the Pregel River, and included two large islands which were connected to each other and the mainland by
Johann Benedict Listing (July 25, 1808 – December 24 1882) was a German mathematician.

Listing was born in Frankfurt and died in Göttingen. In 1847, he wrote a famous article on topology, although he had introduced the term in correspondence some years earlier.
Nature is a prominent scientific journal, first published on 4 November 1869. Although most scientific journals are now highly specialized, Nature is one of the few journals, along with other weekly journals such as Science and