General topology

2020 Mathematics Subject Classification: Primary: 54-XX [MSN][ZBL]

The branch of geometry concerned with the study of continuity and limits at the natural level of generality determined by the nature of these concepts. The initial concepts of general topology are the concepts of a topological space and a continuous mapping, introduced by F. Hausdorff in 1914.

A particular case of a continuous mapping is a homeomorphism — a continuous one-to-one mapping between topological spaces that has a continuous inverse mapping. Spaces that can be mapped onto each other by a homeomorphism (that is, homeomorphic spaces) are regarded as the same in general topology. One of the basic problems in general topology is to find and investigate natural topological invariants — properties of spaces preserved under homeomorphisms (cf. Topological invariant). Of course, any property of a space that can be formulated entirely in terms of its topology is automatically a topological invariant. Proof of the topological invariance of a property of a space is only required when it is formulated with the aid of additional structures defined on the set of points of the space, in some way related to its topology. The topological invariance of the homology groups may serve as an example.

A topological invariant is not necessarily expressible by a number; for example, connectedness, (Hausdorff) compactness and metrizability are topological invariants. Among the numerical invariants (taking numerical values on every topological space), the most important are the dimensional invariants: the small inductive dimension ind, the large inductive dimension Ind and the Lebesgue dimension dim (dimension in the sense of coverings).

Topological invariants of another kind, with cardinal numbers (cf. Cardinal number) as values, play an important role. These cardinal characteristics include the weight of a topological space.

Related to a system of topological invariants there are classes of topological spaces, each class being determined by restriction of one or another topological invariant. The most important classes are metrizable spaces, compact spaces, Tikhonov spaces, paracompact spaces, and feathered spaces (cf. Metrizable space; Compact space; Tikhonov space; Paracompact space; Feathered space).

Fundamental "intrinsic" problems in general topology include: 1) the isolation of important new classes of topological spaces; 2) the comparison of different classes of topological spaces; and 3) the study of spaces within such a class and of categorical properties of this class as a whole. Problem 2) is undoubtedly central in this group, directed to ensuring the internal unity of general topology.

The isolation of important new classes of topological spaces (that is, new topological invariants) is often related to the consideration of additional structures on the space (numerical, algebraic, order), naturally compatible with its topology. Thus, one distinguishes metrizable spaces, ordered spaces, spaces of topological groups, symmetric spaces, etc. The method of coverings plays an important role in solving 1), 2) and 3). In the language of coverings and relations between coverings, the most important of which are the relations of refinement and star refinement, the fundamental classes of compact and paracompact spaces can be singled out, and topological properties like compactness can be formulated. The method of coverings plays an important part in dimension theory.

For the solution of the central problem 2), the method of mutual classification of spaces and mappings is particularly important. It is concerned with establishing connections between various classes of topological spaces by means of continuous mappings subject to certain simple restrictions. Spaces of quite general nature can in this way be described as the images of simpler spaces under "good" mappings. For example, spaces satisfying the first axiom of countability can be characterized as images of metric spaces under continuous open mappings. Connections of this kind establish an effective system of reference in the consideration of classes of topological spaces.

The method of inverse spectra (cf. System (in a category); Spectrum of spaces), closely related to the method of coverings and the method of mappings, enables one to reduce the study of complicated topological spaces to the consideration of systems of mappings of simpler spaces.

Finally, the method of cardinal-valued topological invariants or cardinal functions (cf. Cardinal characteristic) is important in solving 2). Invariants of this kind are very much in accordance with the set-theoretic nature of general topology. In this connection, the system of cardinal-valued invariants has many ramifications and in practice influences all remaining topological properties. Another important feature of cardinal-valued invariants is their close interdependence, on the basis of which rests the possibility to carry out arithmetic operations on such invariants and to compare them by magnitude. Thanks to this feature, the theory of cardinal-valued invariants plays a unifying role in general topology and gives access to each of its sections.

Among the external problems of general topology, the following problem of general character is particularly notable: How are the properties of a topology related to other structures compatible with the topology, and how are they interrelated? Specific problems of this kind are concerned with topological groups, topological vector spaces and measures on topological spaces. To a compact Hausdorff space there corresponds the Banach algebra of all continuous real-valued functions on this space. In this way the theory of topological spaces is closely related to the theory of Banach algebras. Weak topologies (cf. Weak topology) on Banach spaces play a large role in functional analysis. They form a class of non-metrizable topologies important in applications. Every Tikhonov space is uniquely characterized by the ring of all continuous real-valued functions on it in the topology of pointwise convergence. Results of this kind unite general topology and topological algebra.

The concept of a compactification has found applications in potential theory (cf. Boundary (in the theory of uniform algebras); Martin boundary in potential theory).

General topology is important in methodical respects in mathematical education. The fundamental concepts of continuity, convergence and continuous transformation can only be explained and become transparent within the framework of the concepts and constructions of topology. It is hard to name any area of mathematics in which the concepts and language of general topology are not used at all. In particular, its unifying role in mathematics becomes apparent in this. The position of general topology in mathematics is also determined by the fact that a whole series of principles and theorems of general mathematical importance find their natural (i.e. corresponding to the nature of these principles or theorems) formulation only in the framework of general topology. As examples one can mention the concept of compactness — an abstraction from the Heine–Borel lemma on extracting a finite subcovering of an interval, the theorem on the compactness of a product of compact spaces (which generalizes the assertion that a finite-dimensional cube is compact), and the theorem that a continuous real-valued function on a compact space is bounded and attains its least upper and greatest lower bounds. This series of examples could be extended: the concept of a set of the second category, the concept of completeness, the concept of an extension (the very character of these concepts and the results related to them, important for mathematics as a whole, makes their investigation most natural and transparent within the framework of general topology).

#### References[edit]

[1] |  P.S. Aleksandrov, "Theory of functions of a real variable and the theory of topological spaces" , Moscow (1978) pp. 280–358 (In Russian)

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[2] |  P.S. Aleksandrov, "Some results in the theory of topological spaces obtained within the last twenty-five years" Russian Math. Surveys , 15 : 2 (1960) pp. 23–83 Uspekhi Mat. Nauk , 15 : 2 (1960) pp. 25–95
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[a1] |  A.V. Arkhangel'skii, "Function spaces in the topology of pointwise convergence, and compact sets" Russian Math. Surveys , 39 : 5 (1984) pp. 9–56 Uspekhi Mat. Nauk , 39 : 5 (1984) pp. 11–50
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