General topology

The traditional approach to general topology is a set theory . A set is called a topological space when a certain family of its open subsets is given that satisfies the axioms. There are many possible ways to define the structure of a topological space on one set: from discrete to non-Hausdorff “ anti- discrete (trivial) topology, ” gluing all points together.

The basic concepts of set theory ( set , function , ordinal numbers and cardinal numbers , axiom of choice , Zorn's lemma , etc.) are not the subject of a general topology, but are actively used by it. The general topology includes the following sections: properties of topological spaces and their maps, operations on topological spaces and their maps, classification of topological spaces.

In contrast to the differential and algebraic topology , the general topology focuses on the study of the most general form of continuous mappings (topological spaces into each other, and not into spaces endowed with more complex structures: algebraic, etc.). The language of general topology includes such notions as neighborhoods , closures of sets (as well as interiors ), compactness of sets, convergence of sequences and filters .

General topology includes dimensional theory .

General topology originated at the end of the XIX century. and took shape in independent mathematical science at the beginning of the 20th century . The fundamental works belong to F. Hausdorf , A. Poincaré , P. S. Aleksandrov , P. S. Uryson , L. Brauer . In particular, one of the main tasks of the general topology was solved - finding the necessary and sufficient conditions for the metrizability of a topological space.

The most rapid development of general topology as an independent branch of knowledge took place in the middle of the 20th century, at the beginning of the 21st century . rather, it is an auxiliary discipline that “serves” many areas of mathematics: algebraic topology , functional analysis , complex analysis , graph theory, and so on.

• Glossary of general topology

• The concept of the limit of a function, introduced in general topology, allows further generalization in the framework of the theory of pseudotopological spaces .

• P. S. Aleksandrov, V. V. Fedorchuk, V. I. Zaitsev. Highlights in the development of set-theoretic topology
• Alexandrov P. S. Introduction to set theory and general topology - M .: Nauka , 1977
• Arkhangelsky A. V., Ponomarev V. I. Fundamentals of general topology in tasks and exercises - M .: Nauka , 1974
• Bourbaki N. Elements of mathematics. General topology. Basic structures - M .: Science , 1968
• Kelly JL. General topology - M .: Science , 1968
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• Viro O. Ya., Ivanov O.A., Kharlamov V.M., Netsvetaev N. Yu. Elementary topology . Textbook in problems (rus., Eng.)