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# Topology  ( Mathematical Structures )

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In , topology (from the words label=none, and label=none) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as , twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. , and, more generally, are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are and . A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; , which allows distinguishing between a line and a circle; , which allows distinguishing a circle from two non-intersecting circles.

The ideas underlying topology go back to Gottfried Wilhelm Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. 's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century; although, it was not until the first decades of the 20th century that the idea of a topological space was developed.

Motivation
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.

In one of the first papers in topology, demonstrated that it was impossible to find a route through the town of Königsberg (now ) 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 Seven Bridges of Königsberg problem led to the branch of mathematics known as .

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a ." 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 on the sphere. As with the Bridges of Königsberg, 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.

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 . The impossibility of crossing each bridge just once applies to any arrangement of bridges 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 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 progressively enlarging it, while shrinking the hole into a handle.

(1995). 9780387943770, Springer. .

Homeomorphism can be considered the most basic . 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.

History
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 . His 1736 paper on the Seven Bridges of Königsberg 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 edges of a . This led to his polyhedron formula, (where , , and 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.;

Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, and .Richeson (2008) Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print.Listing, Johann Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848 The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated".

Their work was corrected, consolidated and greatly extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as and homology, which are now considered part of algebraic topology.

 + Topological characteristics of closed 2-manifolds none none none none 2 2 2 2

Unifying the work on function spaces of , , Cesare Arzelà, , and others, Maurice Fréchet introduced the 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, coined the term "topological space" and gave the definition for what is now called a .Hausdorff, Felix, "Grundzüge der Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576) Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.

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 as part of his study of . For further developments, see point-set topology and algebraic topology.

The 2022 was awarded to "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".

Concepts

Topologies on sets
The term topology 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 , the , and the can be thought of as the same set with different topologies.

Formally, let be a set and let be a family of subsets of . Then is called a topology on if:

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

If is a topology on , then the pair is called a topological space. The notation may be used to denote a set endowed with the particular topology . By definition, every topology is a .

The members of are called open sets in . A subset of is said to be closed if its complement is in (that is, its complement is open). A subset of may be open, closed, both (a ), or neither. The empty set and itself are always both closed and open. An open subset of which contains a point is called a neighborhood of .

Continuous functions and homeomorphisms
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 to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in . If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to the doughnut.

Manifolds
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 manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an -dimensional manifold has a neighborhood that is to the Euclidean space of dimension . Lines and , but not , are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces, although not all surfaces are manifolds. Examples include the plane, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the and real projective plane, which cannot (that is, all their realizations are surfaces that are not manifolds).

Topics

General topology
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of , called open sets, which is closed under finite and (finite or infinite) . The fundamental concepts of topology, such as continuity, , and , 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 nearby, arbitrarily small, and far apart 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.

are an important class of topological spaces where the distance between any two points is defined by a function called a metric. In a metric space, an open set is a union of open disks, where an open disk of radius centered at is the set of all points whose distance to is less than . Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the , the , real and complex and . Having a metric simplifies many proofs.

Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from to study topological spaces.Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, xii+544 pp. . The basic goal is to find algebraic invariants that classify topological spaces homeomorphism, though usually most classify up to homotopy equivalence.

The most important of these invariants are , homology, and .

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 is again a free group.

Differential topology
Differential topology is the field dealing with differentiable functions on differentiable manifolds.
(2024). 9780387954486, Springer-Verlag.
It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

More specifically, differential topology considers the properties and structures that require only a 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 deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifoldthat is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

Geometric topology
Geometric topology is a branch of topology that primarily focuses on low-dimensional (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.R. B. Sher and R. J. Daverman (2002), Handbook of Geometric Topology, North-Holland. Some examples of topics in geometric topology are orientability, handle decompositions, , crumpling and the planar and higher-dimensional Schönflies theorem.

In high-dimensional topology, characteristic classes are a basic invariant, and is a key theory.

Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive /spherical, zero curvature/flat, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.

2-dimensional topology can be studied as in one variable ( surfaces are complex curves) – by the uniformization theorem every conformal class of metrics 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.

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

Applications

Biology
Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, and have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings. , 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 .
(2024). 9780821836781, American Mathematical Society.
In neuroscience, topological quantities like the Euler characteristic and Betti number have been used to measure the complexity of patterns of activity in neural networks.

Computer science
Topological data analysis 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 ). The main method used by topological data analysis is to:
1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
2. Analyse these topological complexes via algebraic topology – specifically, via the theory of persistent homology.
3. Encode the persistent homology of a data set in the form of a parameterized version of a , which is called a barcode.

Several branches of programming language semantics, such as , are formalized using topology. In this context, Steve Vickers, building on work by and Michael B. Smyth, characterizes topological spaces as Boolean or over open sets, which are characterized as (equivalently, finitely observable) properties.

(1996). 9780521576512, Cambridge University Press.

Physics
Topology is relevant to physics in areas such as condensed matter physics, quantum field theory and physical cosmology.

The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science. Electrical and mechanical properties depend on the arrangement and network structures of and elementary units in materials. The compressive strength of 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 Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics.

A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, , the theory of in algebraic topology, and to the theory of in algebraic geometry. , , , and have all won for work related to topological field theory.

The topological classification of Calabi–Yau manifolds has important implications in , as different manifolds can sustain different kinds of strings.Yau, S. & Nadis, S.; The Shape of Inner Space, Basic Books, 2010.

In cosmology, topology can be used to describe the overall shape of the universe. The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds 2nd ed (Marcel Dekker, 1985, ) This area of research is commonly known as spacetime topology.

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 quantum Hall effect, and then generalized in other areas of physics, for instance in photonics by .

Robotics
The possible positions of a can be described by a called configuration space.John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004 In the area of , one finds paths between two points in configuration space. These paths represent a motion of the robot's and other parts into the desired pose.
(2024). 9783037190548, European Mathematical Society.

Games and puzzles
Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components.. http://sma.epfl.ch/Notes.pdf A Topological Puzzle, Inta Bertuccioni, December 2003. /a> The Figure Eight Puzzle, Science and Math, June 2012.

Fiber art
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 .
(2024). 9781603429733, Storey Publishing.

• Characterizations of the category of topological spaces
• Equivariant topology
• List of algebraic topology topics
• List of examples in general topology
• List of general topology topics
• List of geometric topology topics
• List of topology topics
• Publications in topology
• Topology glossary
• Topological Galois theory
• Topological geometry
• Topological order

Citations

Bibliography

• Ryszard Engelking, General Topology, Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, .
• ; Elements of Mathematics: General Topology, Addison–Wesley (1966).
• (2024). 9780444823755, North Holland.
• (1975). 9780387901251, Springer-Verlag.
• (2024). 9781419627224, Booksurge. .
(Provides a well motivated, geometric account of general topology, and shows the use of groupoids in discussing van Kampen's theorem, , and .)
• Wacław Sierpiński, General Topology, Dover Publications, 2000,
• (2024). 9781560258261, Thunder's Mouth Press. .
(Provides a popular introduction to topology and geometry)

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