# Topology, general

The branch of mathematics whose purpose is to elucidate and investigate ideas of continuity, within the framework of mathematics. Intuitively, the idea of continuity expresses basic properties of space and time, and consequently has a fundamental significance for knowledge. Correspondingly, topology, in which the concept of continuity acquires mathematical substantiation, has naturally penetrated almost all branches of mathematics. In conjunction with algebra, topology forms a general foundation of mathematics, and promotes its unity.

The object of topology is to study those properties of figures, and their mutual disposition, that are preserved under homeomorphisms (cf. Homeomorphism), i.e. one-to-one mappings that are continuous together with their inverses. Consequently, topology can be qualified as a branch of geometry. An important feature of this geometry is the unusual breadth of the class of geometric objects that fall within the sphere of action of its laws.

This breadth is called forth by the fact that the central concept of topology — that of a homeomorphism — does not require in its definition any classical geometrical concepts, such as distance, rectilinearity, linearity, smoothness, etc. The concept of a homeomorphism, and the concept of a continuous mapping lying at its foundation, presuppose only that the points and sets of points of the figure under examination can be in a certain intuitively clear relation of proximity different, generally speaking, from the simple relation of membership.

A "figure" in topology is an arbitrary set of points in which there is given a relation of proximity between points and certain subsets satisfying definite axioms. Such figures are called topological spaces (cf. Topological space). In practice, any figure in the sense of some geometry (affine, projective, differential, etc.) can also be naturally considered as a topological space. In this sense topology is the most general geometry; however, many properties of figures studied in other geometries are consciously ignored in topology.

The main problem in topology is to distinguish and study the topological properties of spaces, or topological invariants (cf. Topological invariant). Among the most important topological invariants are connectivity; compactness; dimension; the weight of a topological space; the fundamental group; and the homology groups (cf. Homology group).

Besides these, a lot of attention is paid to properties of the kind of disposition of one figure in another, or of one topological space in another, that are preserved under homeomorphisms of the ambient space into itself. Problems of this kind began with the Jordan theorem. In the course of development of these ideas the laws of Alexander duality and their generalization, knot theory, were obtained.

In the general approach it is natural to take as central object of research a triple $(X,f,Y)$, where $f$ is a continuous mapping from a topological space $X$ into a topological space $Y$. This embraces the two formulations of the fundamental problem of topology mentioned above. The fundamental tools for comparing triples are, in the spirit of category theory, continuous homomorphisms between them.

A great many topological spaces, or, more properly, types of topological spaces, with which contemporary topology has to deal were described under the influence of various branches of mathematics in response to their very dissimilar requirements. This explains the a priori diversity of the world of topological spaces.

Among the most important classes of topological spaces, formulated from the requirements with which topology is presented by mathematics as a whole, one has in particular: manifolds (smooth, piecewise-linear, topological, etc., cf. Manifold) — locally these topological spaces have the structure of a Euclidean space; polyhedra (cf. Polyhedron, abstract) — these spaces are "pasted together" from elementary figures, like a segment, a triangle, a tetrahedron, etc. (the concept of a simplicial complex, which lies at the basis of the concept of a polyhedron, is an important technical tool for studying polyhedra and manifolds); subspaces of Euclidean spaces (the section of topology dealing with the investigation of these and their disposition of them in space is usually called geometric topology); spaces of functions (such as spaces of continuous functions (mappings) with the topology of pointwise convergence or with the compact-open topology and Banach spaces with the weak topology (cf. Banach space) — topological objects of this kind play a fundamental part in functional analysis and its applications.

A number of important classes of topological spaces are distinguished axiomatically, i.e. by means of isolating some important property of concrete topological objects. Thus, the Heine–Borel theorem (cf. Borel–Lebesgue covering theorem), stating that in every covering of a finite closed interval by open intervals there is a finite subcovering, led to the definition of the abstract concept of compactness (cf. also Compactness, countable) and the class of compact spaces (cf. Compact space) corresponding to it. The presence of natural metrics (cf. Metric) on concrete sets was a starting point for the abstract definition of a metric space and of a metrizable space. The intuitively clear idea of separating points and sets (see Separation axiom) by neighbourhoods was expressed in topology in the definition of the classes of Hausdorff spaces, normal spaces, regular spaces, completely-regular spaces, etc. The class of paracompact spaces, expressing, in particular, the idea of unlimited divisibility of a space, is also important. (Cf. Hausdorff space; Normal space; Completely-regular space; Paracompact space.)

The study of the classes of spaces mentioned above was united by the general idea of a homeomorphism and by the concept of a topological invariant thus generated. Since the concept of a homeomorphism has a clearly expressed set-theoretic nature, set-theoretic methods and constructions of some level of complexity or generality have been used to study each of the classes mentioned above, as well as in the study of other classes of topological spaces. A number of these methods have a general character and are of significance for topology as a whole. At the same time, the study of topological objects within the limits of some fixed class of spaces requires specific, narrower, but also more refined, methods. These methods endow those domains of topology that fall within their sphere of operation with such a clear and distinct character that one sometimes has to speak of the decomposition of topology into a number of independent and loosely related disciplines (e.g. algebraic topology; differential topology; geometric topology; and general topology). However, topology is unified from the outset by its initial concepts and this unity has been confirmed in the process of development of topology by the general value for all branches of topology of a number of fundamental constructions, principles and concepts. Such are the concept of a quotient space, the operation of taking a topological product, the ideas of functional separation, topological approximation and topological extension, principles associated with compactness, etc.

Topological objects formed under the direct influence of other domains of mathematics often have the following important feature: Their topologies are generated by some other, more rigid, mathematical structure naturally associated with the very nature of the object under investigation. In connection with this there arise the following related problems of a fundamental nature:

1) How are the invariants of a given "exterior" structure (combinatorial, differential, algebraic, etc.) associated with the topological invariants of the topology generated by this structure?

2) Which invariants of the given "exterior" structure are invariants of the topology generated by it, i.e. which are topological invariants?

3) How many, up to isomorphism, different "exterior" structures of a given type and generated by a given topology exist? First of all, it is important to clarify whether there is at least one such structure, and of special significance is the case when this structure is unique (up to isomorphism); it is then itself a (topological) invariant of the topology under consideration (hence so are all its characteristics).

These general questions acquire an important concrete content, for example, in the topology of manifolds.

In principle, problems on the relations between metric and topological invariants and on the existence of a metric defining the given topology (the metrization problem) are of a similar character.

In the case of more general spaces there arises the problem of the relation between the invariants of the uniform structure and the topological structure generated by it. The study of uniform invariants and their relations with topological invariants is the object of uniform topology (cf. Uniform space). Another structure closely associated with the topology is that of proximity. The concept of a proximity space is based on a proximity relation between the subsets of the space, in distinction from the concept of a topological space. The character of the statement of the basic problem in topology changes, depending on the size of the class of topological spaces under consideration. Thus, after restricting to a narrow class of spaces, one is faced with the problem of distinguishing them from each other, up to homeomorphism, in terms of topological invariants. This problem looks completely natural, for example, within the class of topological manifolds, but even there it proves to be very difficult and it is probably algorithmically unsolvable. The complexity of the problem of distinguishing manifolds up to a homeomorphism leads to the necessity of considering the relation of homotopy equivalence of topological spaces (which is coarser than distinguishing up to homeomorphism). At the basis of this relation lies the concept of a homotopy of one continuous mapping into another, which is of a pure set-theoretic character.

Although methods of algebraic topology play an exceptionally important role in topological research, purely set-theoretic constructions also play an essential part. This is associated with the fact that, for example, the relation of homotopy equivalence applied to manifolds leads outside the class of manifolds. Here one obtains simpler topological objects, the study of which is very useful in technical respects. Methods of homotopy theory require the realization of set-theoretic constructions, like various kinds of "sweeping-out" , glueing one topological space to another along an arbitrary continuous mapping, etc. This leads to the concepts of a CW-complex and a cellular space; the latter also form the maximal class of spaces including all differentiable manifolds and polyhedra, and admit a sufficiently complete study by methods of algebraic topology.

For broader classes of spaces, such as the class of all compacta, the class of all paracompacta or the class of all metrizable spaces, posing the problem of distinguishing such spaces up to a homeomorphism by means of an extensive system of computable topological invariants is not possible because of its intuitive unsolvability. The basic problem of topology here is the problem of comparing not individual topological spaces but whole classes of topological spaces which, especially in an axiomatic approach, usually correspond to distinct topological invariants or combinations of them. Under such an approach the basic problem of topology is transformed into the problem of the systematic comparison of topological invariants. In this way a systematic and developed classification of topological spaces has been successfully constructed.

Two methods predominate in the solution of this problem. First, there is the method of mutual classification of spaces and mappings. This is the study of the behaviour of topological invariants under different kinds of continuous mappings and also the study of when a topological space from a given class can be represented as the image of a space from another given class under a continuous mapping of one kind or another. This problem is all the more important and natural because often topological spaces are given and are already related by certain continuous mappings; e.g., when a new space is constructed as the quotient space of a certain initial topological space.

The second method of comparison is the application of cardinal, or cardinal-valued, topological invariants, also called cardinal characteristics, cf. Cardinal characteristic. Since infinite cardinal numbers are the values of cardinal invariants, this gives a possibility of comparing them by using operations and laws on cardinal numbers. This direction of topology depends on deep statements of set theory, such as Martin's axiom and the continuum hypothesis. Suslin's conjecture (the unsolvability of which within the framework of the Zermelo–Fraenkel system of axioms of set theory has been proved) can be formulated in the language of cardinal invariants. The following is a characteristic argument with cardinal invariants: For metrizable spaces the density and weight coincide; hence if the weight and the density are different for a given space, the space is not metrizable. In the theory of cardinal invariants many subtle and unexpected results have been obtained.

In spite of the specific nature, mentioned above, acquired by topological problems and methods depending on the class of topological spaces that is selected for study, a number of basic problems determining the development of topology have been formulated in a general manner for all its branches and have been solved by some general principles and methods.

The following problems are examples of these.

a) The construction of a system of topological invariants on the basis of a topology or of exterior structures generating it. In this case there arises the problem of finding these invariants for individual spaces and classes of spaces.

b) The study of the behaviour of topological invariants under basic operations on topological spaces, in particular, under transition to a subspace.

c) The study of the behaviour of topological invariants under different kinds of continuous mappings (in particular, under imbeddings).

d) The study of the relations between the topological properties of spaces and their complements in a certain ambient space. Many results of geometric topology, duality theorems, and results relating properties of topological spaces and their remainders in compact Hausdorff extensions, are good illustrations of this direction.

Among the general methods that can be applied to solve the majority of problems of topology in all its branches are:

$\alpha$) The method of coverings (cf. Covering (of a set)). This method gives a result in the solution of metrization problems, in the determination and study of paracompact spaces and in the determination and study of fundamental objects of combinatorial topology (simplicial and cellular complexes). The approximation of topological spaces by polyhedra is based on the method of coverings, in particular, on the concept of the nerve of a covering (cf. Nerve of a family of sets). Theorems on the immersion of manifolds in a Euclidean space are proved using open coverings and partitions of unity corresponding to them.

$\beta$) The method of functors (cf. Functor). This consists in relating algebraic and algebraic-topological objects with correct (functorial) behaviour and admitting calculation for topological spaces. The homology group, the cohomology ring, and the $K$-functor, associated with the concept of a vector bundle over a topological space, generalizing the concept of a tangent manifold, are important examples of functors. The algebraic method in topology is based on the use of such functors.

$\gamma$) The method of spectra (cf. Spectrum of spaces). Its essence is in the representation of spaces with a highly complex structure as limits of inverse spectra of simpler spaces (e.g. polyhedra). In this connection one studies the relation between topological invariants of elements of the spectrum and the limit space. The concept of a spectrum realizes in some form the idea of topological approximation of a topological space by objects that are simpler or more convenient for study.

The construction of a cohomology theory for wide classes of spaces and the construction of examples of complex topological spaces with given combinations of properties are based on this method.

$\delta$) The method of continuous mappings: imbeddings, mappings of spaces from one class onto spaces of another class (cf. Continuous mapping). Here, the study of the behaviour of topological invariants constitutes the essence of this method. An important part is played by the solution of the problem of the continuous extension of a mapping defined on a part of a space to the whole space. The solution to this problem essentially depends on the homotopy properties of the space, and it occupies a central place in homotopy theory. Associated specifically with this problem are obstruction theory and the theory of retracts (cf. Obstruction; Retract of a topological space).

$\epsilon$) The axiomatic method. This method gives the widest and most natural framework for the elucidation of mutual relations between topological invariants and for the definition of new topological invariants and classes of topological spaces "within" topology itself, in agreement with the necessity of making this classification systematic and harmonious. Here one fixes a topological invariant, having defined it in terms of the topology itself and usually abstracting from concrete ways of presenting the spaces of the class under consideration, and disregards the problem of a method for calculating this topological invariant. Thus arise the class of compacta, the class of continua, etc.

The applications of topology have a dual character, determined by which branch of topology is applied and where it is applied. Clearly, applications of topology are possible wherever the idea of continuity exists.

In spite of the very diverse applications of topology in concrete situations, which result from the statements expressed above, one may indicate a number of general principles and concepts on which these applications are most often based. Thus, the theory of manifolds has most direct applications in mechanics and in the theory of differential equations; homology theory has extended beyond the framework of topology and has developed into an important independent discipline — homological algebra, which has been applied in algebraic geometry, the theory of Banach algebras, etc. The following concepts acquired algebraic treatment and obtained applications associated with this: a manifold; the $K$-functor, arising from differential topology; cobordism theory, which is important in the development of differential topology and has been applied in algebraic geometry (the Riemann–Roch theorem), in the theory of elliptic operators (index formulas), etc. The degree of a mapping is also important in applications. A proof of the so-called fundamental theorem of algebra is based on it. The application of methods and concepts in homology and homotopy to infinite-dimensional function spaces had an essential influence on analysis — in particular, in connection with theorems on the existence of solutions of partial differential equations. Fixed-point theorems for continuous mappings also have important applications. These theorems are of a mixed set-theoretic and algebraic nature, and their applications have a qualitative character; they are aimed not at the calculation of some objects or other, but at proving their existence. A number of important principles combining topological and linear structures have the same purpose. Such are, e.g., the Krein–Mil'man theorem on extreme points of a convex compactum, the Banach–Steinhaus theorem, the closed-graph theorem, Alaoglu's theorem on the compactness of the unit ball in the weak topology, the Eberlein–Shmul'yan theorem on compacta in Banach spaces with the weak topology, etc.

There are a number of topological principles and concepts of a "pure" set-theoretic character. Among these: the concept of compactness (countable compactness); the Tikhonov theorem on the compactness of the topological product of compact spaces; the theorem on the closedness of a compact set in an arbitrary Hausdorff space; the characterization of compactness as absolute closedness; the Stone–Weierstrass theorem; completeness and the principles related to it; the fixed-point theorem for a contraction mapping; the Baire theorem on the non-emptiness of the intersection of a countable family of everywhere-dense open sets; etc. The topological dimension, together with compactness and completeness, is no doubt one of the most important general mathematical concepts.

In a number of constructions in functional analysis, potential theory, etc. the concepts of an extension of a topological space and of a boundary play an essential role (in the algebra of functions: the Shilov boundary, the Martin boundary and the Choquet boundary).

The nature of topological dynamics requires a rather extensive abstraction of set-theoretic concepts and constructions of topology. Only this will give the natural framework for the discussion and analysis of such concepts as the limit set of a trajectory, almost periodicity, a minimal set, Lagrange stability, Poisson stability, a non-wandering point, etc. Once again, compactness plays a very important part here.

The concepts and methods of topology, especially the set-theoretic ones, naturally occur in topological algebra. In applying topological methods one must bear in mind that in the presence of some algebraic structure that is compatible with the topology the relations between the topological invariants may change strongly: many familiar relations are simplified and new profound relations appear.

Set-theoretic constructions of topology have important applications in mathematical logic.

How to Cite This Entry:
Topology, general. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topology,_general&oldid=42992
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article