Topology

Topology (Greek τόπος tópos, German 'Ort, Platz' and -logy) is a fundamental branch of mathematics. It deals with the properties of mathematical structures that are conserved under continuous deformations, the notion of continuity being defined by topology in a very general way. Topology emerged from the concepts of geometry and set theory.

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

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

The fundamental notion of topology is that of topological space, which is a far-reaching abstraction of the notion of "closeness" and thus allows far-reaching generalizations of mathematical concepts such as continuity and limit. Many mathematical structures can be conceived of as topological spaces. Topological properties of a structure are called those that depend only on the structure of the underlying topological space. These are just those properties which are not changed by "deformations" or by homeomorphisms. In illustrative cases, these include the stretching, compressing, bending, distorting, and twisting of a geometric figure. For example, a sphere and a cube are indistinguishable from the point of view of topology; they are homeomorphic. Similarly, a donut (whose shape is called a solid torus in mathematics) and a one-handled cup are homeomorphic because one can be transformed into the other without cutting (see animation at right). In contrast, the surface of the torus is topologically distinct from the surface of the sphere: On the sphere, any closed curve can be steadily contracted to a point (the descriptive language can be made more precise); on the torus, not any.

Topology is divided into subfields. These include algebraic topology, geometric topology, and topological graph and knot theory. Set-theoretic topology can be regarded as the basis for all these subdisciplines. In particular, topological spaces whose properties differ particularly widely from those of geometric figures are also considered in it.

An important concept in topology is continuity. Continuous mappings correspond in 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 an isomorphism in other categories: homeomorphic spaces are indistinguishable by topological means. A fundamental problem of this discipline is to decide whether two spaces are homeomorphic, or more generally, whether continuous mappings with certain properties exist.

Cup and solid torus are homeomorphic to each other. Remark: A homeomorphism is a direct mapping between the points of the cup and the solid torus, the intermediate steps in the temporal course serve only to illustrate the continuity of this mapping. .

History

The term "topology" is first found around 1840 by Johann Benedict Listing; however, the older term analysis situs (roughly 'positional investigation') remained in common use for a long time, with a focus of meaning beyond the more recent, "set-theoretic" topology.

The solution of the seven-bridge problem of 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 of Euler to the so-called analysis situs is the polyhedron theorem of 1750 named after him. If one denotes with the number of vertices, with that of edges and with that of faces of a polyhedron (which still satisfies conditions to be specified), then e-k+f=2 . It was not until 1860, when a copy (made by Gottfried Wilhelm Leibniz) of a lost manuscript by René Descartes became known, that the latter had already known the formula.

Maurice Fréchet introduced metric space in 1906. Georg Cantor dealt with the properties of open and closed intervals, investigated boundary processes, and at the same time founded modern topology and set theory. Topology is the first branch of mathematics to be consistently formulated in terms of set theory - and, conversely, gave impetus to 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 there an open ambient basis, but not a topology, which was introduced only by Kazimierz Kuratowski and Heinrich Tietze, respectively, around 1922. The axioms were then popularized in this form by the textbooks of Kuratowski (1933), Alexandroff/Hopf (1935), Bourbaki (1940), and Kelley (1955). It turned out that many mathematical insights could be applied to this conceptual basis. For example, it was recognized that for a fixed basic set there exist different metrics that led to the same topological structure on that set, but also that different topologies are possible on the same basic set. On this basis, set-theoretic topology developed into an independent field of research, which in a certain sense spun off from geometry - or rather, is closer to calculus than to geometry proper.

One goal of topology is to develop invariants of topological spaces. These invariants can be used to distinguish topological spaces. For example, the gender of a compact, connected orientable surface is one such invariant. The sphere with gender zero and the torus with gender one are distinct topological spaces. Algebraic topology arose from thoughts of Henri Poincaré on the fundamental group, which is also an invariant in topology. Over time, topological invariants such as the Betti numbers studied by Henri Poincaré were replaced by algebraic objects such as homology and cohomology groups.

Basic Terms

Topological space

→ Main article: Topological space

Topology (as a branch of mathematics) deals with properties of topological spaces. If an arbitrary 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 space is then determined by the fact that certain subsets are distinguished as open. The identical topological structure can be specified in terms of their complements, but these then represent the closed subsets. Usually, topological spaces are defined in textbooks in terms of the open sets; more precisely, the set ${\mathcal {O}}$ open sets is called the topology of the topological space . $(X,\mathcal{O})$

Starting from open or closed sets, respectively, numerous topological notions can be defined, such as those of the neighborhood, continuity, touch point, and convergence.

Open quantities

→ Main article: Open set

Topology (over open sets): a topological space is a set of points provided with a set ${\mathcal {O}}\subset {\mathcal {P}}\left(X\right)$ of subsets (the open sets) satisfying the following conditions:

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

We call the pair $(X, \mathcal{O})$ a topological space and ${\mathcal {O}}$ the topology of this topological space.

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

$f\colon X\to Y$

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

Closed quantities

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

This immediately results in the

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

$X\in {\mathcal {A}}$ and $\emptyset \in {\mathcal {A}}$ .
For arbitrary index sets with $A_{i}\in {\mathcal {A}}$ for all $i\in I$ holds.	$\textstyle \bigcap _{i\in I}A_{i}\in {\mathcal {A}}$ .	(average)
For finite index sets with $A_{i}\in {\mathcal {A}}$ for all $i\in I$ holds.	$\textstyle \bigcup _{i\in I}A_{i}\in {\mathcal {A}}$ .	(Association)

The equivalence to the previous definition over open sets follows immediately from De Morgan's laws: from $\textstyle \bigcap$ becomes $\textstyle \bigcup$ and vice versa.

Closed sets can be thought of as sets of points that contain their edge, or in other words, whenever there are points of the closed set that are arbitrarily close to another point (a touch point), that point is also contained in the closed set. One considers what basic properties should be contained in the notion of closed set and then, abstracting from specific definitions of closedness, such as from calculus, calls any set provided with closed subsets (satisfying these conditions) a topological space. First of all, the empty set should be closed, because it does not contain any points that could touch others. Similarly, the set of all points should be closed, because it already contains all possible touch points. If any set of closed sets is given, then the intersection, that is, the set of points contained in all these sets, should also be closed, for if the intersection had touch points lying outside of it, then already one of the sets to be intersected would not have to contain this touch point, and could not be closed. Moreover, the union of two (or finitely many) closed sets is again said to be closed; thus, in the union of two closed sets, no touchpoints are added. The union of infinitely many closed sets, on the other hand, is not required to be closed, because they could "keep approaching" another point and thus touch it.

Further definitions

→ Main article: Axiom systems of general topology

Homeomorphism

→ Main article: Homeomorphism

A homeomorphism is a bijective mapping between two topological spaces, such that by pointwise transferring the open sets, a bijection between the topologies of the two spaces also comes about; in the process, each open set must be mapped to an open set. Two topological spaces between which there is a homeomorphism are called homeomorphic. Homeomorphic spaces do not differ with respect to topological properties in the strict sense. The homeomorphisms can be taken 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 one investigates uniform spaces, metric spaces, topological groups or topological algebras. Properties which rely on such additional structures are no longer necessarily preserved under homeomorphisms, but are also partly the object of investigation of various subfields of topology.

Generalizations of the concept of topological space also exist: In point-free topology, instead of a set of points with sets labeled as open, one considers only the structure of the open sets as a lattice. Convergence structures define against which values each filter converges on an underlying set of points. Under the catchword Convenient Topology one tries to find classes of spaces similar to topological or uniform spaces, but with more "pleasant" category-theoretic properties.