# Topology (mathematics)

Cup and full torus are homeomorphic to one another.
Note : A homeomorphism is a direct mapping between the points of the cup and the full torus, the intermediate stages in the course of time only serve to illustrate the continuity of this mapping.

The topology ( Greek τόπος tópos , German , location, place ' and -logie ) is a fundamental part of mathematics . It deals with the properties of mathematical structures that are preserved under constant deformations, whereby the concept of continuity is defined in a very general way by the topology. The topology emerged from the concepts of geometry and set theory .

Towards the end of the 19th century, topology emerged as a separate discipline, called geometria situs 'geometry of location' or analysis situs (Greek-Latin for 'analysis of location') in Latin.

Topology has been recognized as a fundamental discipline for decades. Accordingly, alongside algebra , it can be seen as a second pillar for a large number of other fields of mathematics. It is particularly important for geometry , analysis , functional analysis and the theory of Lie groups . For its part, it has also fertilized set theory and category theory .

The basic concept of topology is that of topological space , which represents a far-reaching abstraction of the notion of “proximity” and thus allows far-reaching generalizations of mathematical concepts such as continuity and limit value . Many mathematical structures can be understood as topological spaces. Topological properties of a structure are called those that only depend on the structure of the underlying topological space. These are precisely those properties that are not changed by “deformations” or by homeomorphisms . In illustrative cases, this includes stretching, compressing, bending, distorting and twisting a geometric figure. For example, a sphere and a cube are indistinguishable from a topology perspective; they are homeomorphic. Likewise, a donut (the shape of which is called a full torus in mathematics ) and a one-handled cup are homeomorphic, as one can be transformed into the other without cutting (see animation on the right). In contrast, the surface of the torus is topologically different from the spherical surface: on the sphere, every closed curve can be continuously drawn together to a point (the descriptive language can be made more precise), on the torus not every curve .

The topology is divided into sub-areas. These include the algebraic topology , the geometric topology as well as the topological graph and knot theory . The set theoretical topology can be seen as the basis for all of these sub-disciplines. In this, topological spaces in particular are also considered, the properties of which differ particularly widely from those of geometric figures.

An important concept in topology is continuity. Continuous mappings correspond in the topology to what is usually called homomorphisms in other mathematical categories . A reversible mapping between topological spaces that is continuous in both directions is called a homeomorphism and corresponds to what is usually called isomorphism in other categories: Homeomorphic spaces cannot be distinguished by topological means. A fundamental problem in this discipline is to decide whether two spaces are homeomorphic or, more generally, whether continuous mappings with certain properties exist.

## history

The term “topology” was first used around 1840 by Johann Benedict Listing ; the older term analysis situs (for example, situation investigation ”) remained common for a long time, with a meaning that went beyond the newer,“ set-theoretical ”topology.

The solution of the seven bridges problem at Königsberg by Leonhard Euler in 1736 is considered to be the first topological and at the same time the first graph theoretical work in the history of mathematics. Another contribution from Euler to the so-called Analysis situs is the polyhedron set of 1750, named after him. If one denotes the number of vertices with which the edges and with those of the surfaces of a polyhedron (which meets the conditions to be specified), then applies . It was not until 1860 that it became known through a copy (made by Gottfried Wilhelm Leibniz ) of a lost manuscript by René Descartes that he had already known the formula. ${\ displaystyle e}$${\ displaystyle k}$${\ displaystyle f}$${\ displaystyle e-k + f = 2}$

Maurice Fréchet introduced the metric space in 1906 . Georg Cantor dealt with the properties of open and closed intervals, examined boundary processes, and at the same time founded modern topology and set theory . Topology is the first branch of mathematics that was consistently formulated in terms of set theory - and, conversely, gave impetus for the development of set theory.

A definition of topological space was first established by Felix Hausdorff in 1914. According to today's usage, he defined an open environment base there , but not a topology that was only introduced by Kazimierz Kuratowski and Heinrich Tietze around 1922. In this form, the axioms were popularized in the textbooks by Kuratowski (1933), Alexandroff / Hopf (1935), Bourbaki (1940) and Kelley (1955). It turned out that a lot of mathematical knowledge can be transferred to this conceptual basis. For example, it was recognized that there are different metrics for a fixed base set, which lead to the same topological structure on this set, but also that different topologies are possible on the same base set. On this basis, the set-theoretical topology developed into an independent research area, which in a certain way has been separated from geometry - or rather is closer to analysis than to actual geometry.

One goal of topology is the development of invariants of topological spaces. With these invariants, topological spaces can be distinguished. For example, the gender of a compact , coherent, orientable surface is such an invariant. The gender zero sphere and the gender one torus are different topological spaces. The algebraic topology emerged from the considerations of Henri Poincaré about the fundamental group , which is also an invariant in the topology. Over time, topological invariants such as the Betti numbers studied by Henri Poincaré have been replaced by algebraic objects such as homology and cohomology groups .

## Basic concepts

### Topological space

Topology (as a branch of mathematics ) deals with properties of topological spaces . If any basic set is provided with a topology (a topological structure), then it is a topological space and its elements are understood as points . The topology of the room is then determined by the fact that certain subsets are marked as open . The identical topological structure can be specified via its complements , which then represent the closed subsets. Usually, topological spaces are defined in textbooks using the open sets; more precisely: the set of open sets is called the topology of topological space . ${\ displaystyle {\ mathcal {O}}}$${\ displaystyle (X, {\ mathcal {O}})}$

Based on open or closed sets, numerous topological terms can be defined, such as those of the environment , continuity , point of contact and convergence .

#### Open sets

Topology (via open sets): A topological space is a set of points provided with a set of subsets (the open sets), which meets the following conditions: ${\ displaystyle X}$${\ displaystyle {\ mathcal {O}} \ subset {\ mathcal {P}} \ left (X \ right)}$

 ${\ displaystyle X \ in {\ mathcal {O}}}$and .${\ displaystyle \ emptyset \ in {\ mathcal {O}}}$ For any index sets with for all applies${\ displaystyle I}$${\ displaystyle O_ {i} \ in {\ mathcal {O}}}$${\ displaystyle i \ in I}$ ${\ displaystyle \ textstyle \ bigcup _ {i \ in I} O_ {i} \ in {\ mathcal {O}}}$. (Union) For finite index sets with for all applies${\ displaystyle I}$${\ displaystyle O_ {i} \ in {\ mathcal {O}}}$${\ displaystyle i \ in I}$ ${\ displaystyle \ textstyle \ bigcap _ {i \ in I} O_ {i} \ in {\ mathcal {O}}}$. (Average)

The pair is called a topological space and the topology of this topological space. ${\ displaystyle (X, {\ mathcal {O}})}$${\ displaystyle {\ mathcal {O}}}$

The most important concept that is defined by open sets is that of the environment: A set is the environment of a point if it includes an open set that contains the point. Another important concept is that of continuity : a mapping

${\ displaystyle f \ colon X \ to Y}$

of the topological spaces and is continuous if and only if the archetypes of open sets are open in , so it holds. ${\ displaystyle (X, T_ {X})}$${\ displaystyle (Y, T_ {Y})}$${\ displaystyle f ^ {- 1} (O_ {Y})}$${\ displaystyle O_ {Y} \ in T_ {Y}}$${\ displaystyle (X, T_ {X})}$${\ displaystyle f ^ {- 1} (O_ {Y}) \ in T_ {X}}$

#### Closed sets

Starting from the open sets, the closed sets can be defined as those subsets of space whose complements are open, i.e. for every open set the points that are not contained in it form a closed set. ${\ displaystyle O}$${\ displaystyle A: = X \! \ setminus \! O}$

This immediately results in the

Topology (over closed sets): A topological space is a set of points provided with a set of subsets of (the closed sets; is the power set of ), which satisfies the following conditions: ${\ displaystyle X}$${\ displaystyle {\ mathcal {A}} \ subset {\ mathcal {P}} \ left (X \ right)}$${\ displaystyle X}$${\ displaystyle {\ mathcal {P}} \ left (X \ right)}$${\ displaystyle X}$

 ${\ displaystyle X \ in {\ mathcal {A}}}$and .${\ displaystyle \ emptyset \ in {\ mathcal {A}}}$ For any index sets with for all applies${\ displaystyle I}$${\ displaystyle A_ {i} \ in {\ mathcal {A}}}$${\ displaystyle i \ in I}$ ${\ displaystyle \ textstyle \ bigcap _ {i \ in I} A_ {i} \ in {\ mathcal {A}}}$. (Average) For finite index sets with for all applies${\ displaystyle I}$${\ displaystyle A_ {i} \ in {\ mathcal {A}}}$${\ displaystyle i \ in I}$ ${\ displaystyle \ textstyle \ bigcup _ {i \ in I} A_ {i} \ in {\ mathcal {A}}}$. (Union)

The equivalence to the previous definition of open sets follows directly from De Morgan's laws : from becomes and vice versa. ${\ displaystyle \ textstyle \ bigcap}$${\ displaystyle \ textstyle \ bigcup}$

Closed sets can be imagined as sets of points that contain their edge, or in other words: Whenever there are points in the closed set that come as close as desired to another point (a point of contact ), this point is also included in the closed set . One thinks about which basic properties should be contained in the concept of the closed set and then , abstracting from specific definitions of closedness, e.g. from analysis , calls every closed subsets (which meet these conditions) a topological space. First of all, the empty set should be closed, because it does not contain any points that others could touch . Likewise, the set of all points should be completed, because it already contains all possible points of contact. If any set of closed sets is given, the intersection, i.e. the set of points that are contained in all these sets, should also be closed, because if the intersection had contact points that lie outside of it, one of the would have to be cutting quantities do not contain this point of contact and could not be final. In addition, the union of two (or finitely many) closed sets should again be closed; when two closed sets are united, there are no additional points of contact. The union of an infinite number of closed sets, on the other hand, is not required to be closed, because these could “keep getting closer” to another point and thus touch it.

### Homeomorphism

A homeomorphism is a bijective mapping between two topological spaces, so that a bijection between the topologies of the two spaces also comes about through point-by-point transfer of the open sets. Two topological spaces between which there is a homeomorphism are called homeomorphic . Homeomorphic spaces do not differ in terms of topological properties in the narrower sense. The homeomorphisms can be understood as the isomorphisms in the category of topological spaces.

### Terms not related to topological spaces

Topological spaces can be equipped with additional structures, for example uniform spaces , metric spaces , topological groups or topological algebras are examined . Properties that make use of additional structures of this kind are no longer necessarily preserved under homeomorphisms, but are sometimes also the subject of investigation in various sub-areas of topology.

There are also generalizations of the concept of topological space: In point-free topology , instead of a set of points with sets marked as open, only the structure of the open sets is considered as a lattice . Convergence structures define to which values ​​each filter converges on an underlying set of points. Under the catchphrase Convenient Topology , an attempt is made to find classes of spaces similar to topological or uniform spaces, but which have “more pleasant” category-theoretical properties.

## Sub-areas of the topology

The modern topology is roughly divided into the three areas of set-theoretical topology, algebraic topology and geometric topology. There is also the differential topology . This is the basis of modern differential geometry and, despite the extensively used topological methods, is mostly regarded as a sub-area of ​​differential geometry.

### Set theoretical or general topology

The set theoretical topology, like the other sub-areas of topology, includes the study of topological spaces and the continuous mappings between them. Especially those for the Analysis fundamental concepts of continuity and convergence are completely transparent only in the terminology of set-theoretic topology. But the concepts of set theoretic topology are also used in many other mathematical sub-areas. In addition, there are many concepts and mathematical statements of the set-theoretical topology that are valid and important for the more specific sub-areas of topology. Examples:

For example, the compactness of a room is an abstraction of the Heine – Borel principle . In the general terminology of set-theoretical topology, the product of two compact spaces is compact again, which generalizes the statement that a closed finite-dimensional cube is compact. In addition, it holds that a continuous function is bounded by a compact set into the real numbers and takes on its maximum and minimum. This is a generalization of the principle of minimum and maximum .

In general, topological spaces can violate many properties that are familiar from the topology of real numbers, but which are often found in normal spaces. Therefore, one often looks at topological spaces that meet certain separation properties, which represent minimal requirements for many more extensive sentences and enable more in-depth characterizations of the structure of the spaces. Compactness is another example of such “beneficial” properties. In addition, one also considers spaces on which certain additional structures are defined, such as uniform spaces or even topological groups and metric spaces , which through their structure allow additional concepts such as completeness .

Another key concept in this sub-area is different concepts of context .

### Algebraic topology

The algebraic topology (also called "combinatorial topology", especially in older publications) examines questions about topological spaces by tracing the problems back to questions in algebra . In algebra, these questions are often easier to answer. A central problem within topology is, for example, the investigation of topological spaces for invariants . Using the theory of homologies and cohomologies , one searches for such invariants in the algebraic topology.

### Geometric topology

The geometric topology deals with two-, three- and four-dimensional manifolds . The term two-dimensional manifold means the same as surface, and three- and four-dimensional manifolds are corresponding generalizations. In the field of geometric topology, one is interested in how manifolds behave under continuous transformations. Typical geometric quantities such as angle, length and curvature vary under continuous images. A geometrical quantity that does not vary, and therefore one that is of interest, is the number of holes in a surface. Since one deals almost exclusively with manifolds with dimensions smaller than five, this sub-area of ​​topology is also called low-dimensional topology. In addition, the knot theory as part of the theory of three-dimensional manifolds belongs to the geometric topology.

## Applications

Since the field of topology is very broad, aspects of it can be found in almost every branch of mathematics. The study of the respective topology therefore often forms an integral part of a deeper theory. Topological methods and concepts have become an integral part of mathematics. A few examples are given here:

Differential geometry

Manifolds

In differential geometry , the study of plays manifolds a central role. These are special topological spaces , i. H. Sets that have a certain topological structure. Often they are also called topological manifolds. Fundamental properties are then proven with the help of topological means before they are provided with further structures and then form independent (and non-equivalent) subclasses (e.g. differentiable manifolds , PL manifolds, etc.).

Example used result of the geometric topology: Classification of surfaces

Closed surfaces are special types of 2-dimensional manifolds. With the help of the algebraic topology it can be shown that every surface consists of a finite number of embedded 2-polytopes that are glued together along their edges. In particular, this allows all closed areas to be classified into 3 classes, which is why one can always assume that the closed area is in a "normal form".

Functional Analysis The functional analysis arose from the study of function spaces which initially abstractions as Banach and Hilbert spaces learned. Today, functional analysis also deals more generally with infinite-dimensional topological vector spaces . These are vector spaces provided with a topology so that the basic algebraic operations of the vector space are continuous (“compatible” with the topology). Many of the concepts examined in functional analysis can be traced back solely to the structure of topological vector spaces, as which Hilbert and Banach spaces in particular can be understood, so that they can be viewed as the central object of investigation in functional analysis.

Descriptive Set Theory The descriptive set theory deals with certain "constructible" and "well-formed" subsets of Polish spaces . Polish spaces are special topological spaces (without any further structure) and many of the central concepts examined are purely topological in nature. These topological terms are related to concepts of “ definability ” and “ predictability ” from mathematical logic , about which statements can be made using topological methods.

Harmonic Analysis The central object of investigation in harmonic analysis are locally compact groups , that is, groups provided with a compatible locally compact topological structure. These represent a generalization of the Lie groups and thus of ideas of "continuous symmetries".

### Application in economics

Topological concepts are mainly used in economics in the field of welfare economics . The topology is also used in general equilibrium models .

## literature

### Textbooks

Commons : Topology (Mathematics)  - album of pictures, videos and audio files
Wikibooks: Mathematics: Topology  - Learning and Teaching Materials

## Individual evidence

1. ^ IM Jones (ed.): History of Topology . Elsevier, 1999, ISBN 0-444-82375-1 , pp. 103 .
2. ^ IM Jones (ed.): History of Topology . Elsevier, 1999, ISBN 0-444-82375-1 , pp. 503-504 .
3. Christoph J. Scriba, Peter Schreiber: 5000 years of geometry: history, cultures, people (from counting stone to computer). Springer, Berlin, Heidelberg, New York, ISBN 3-540-67924-3 , p. 451.
4. a b c F. Lemmermeyer: Topology . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .
5. Felix Hausdorff: Basic features of set theory , 1914, p. 213.
6. ^ Find. Math. , 3 , 1922.
7. ^ Math. Ann. 88, 1923.
8. Epple et al., Hausdorff GW II, 2002.
9. Christoph J. Scriba, Peter Schreiber: 5000 years of geometry: history, cultures, people (from counting stone to computer). Springer, Berlin, Heidelberg, New York, ISBN 3-540-67924-3 , p. 515.
10. If a topological space, then is the set of closed sets ${\ displaystyle (X, {\ mathcal {O}})}$
${\ displaystyle {\ mathcal {A}}: = \ {A \ in {\ mathcal {P}} (X) \; \ mid \; X \! \ setminus \! A \ in {\ mathcal {O}} \}}$.
11. General topology . In: Michiel Hazewinkel (Ed.): Encyclopedia of Mathematics . Springer-Verlag and EMS Press, Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
12. John. Stillwell: Mathematics and its history . Springer, New York 2010, ISBN 978-1-4419-6052-8 , pp. 468 .
13. D. Erle: Knot theory . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
14. Berthold U. Wigger: Grundzüge der Finanzwissenschaft , p. 18.