- General topology
In

mathematics ,**general topology**or**point-set topology**is the branch oftopology which studies properties oftopological space s and structures defined on them. It is distinct from other branches of topology in that the topological spaces may be very general, and do not have to be at all similar tomanifold s.**History**General topology grew out of a number of areas, most importantly the following:

*the detailed study of subsets of the

real line (once known as the "topology of point sets", this usage is now obsolete)

*the introduction of themanifold concept

*the study ofmetric space s, esp.normed linear space s, in the early days offunctional analysis General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.

**Scope**More specifically, it is in general topology that basic notions are defined and theorems about them proved. This includes the following:

* open and

closed set s;

* interior and closure;

* neighbourhood and closeness;

* compactness and connectedness;

* continuous functions;

* convergence ofsequence s, nets, and filters;

*separation axiom s

* countability axiomOther more advanced notions also appear, but are usually related directly to these fundamental concepts, without reference to other branches of mathematics.

Set-theoretic topology examines such questions when they have substantial relations toset theory , as is often the case.Other main branches of topology are

algebraic topology ,geometric topology , anddifferential topology . As the name implies, general topology provides the common foundation for these areas.An important variant of general topology is

pointless topology , which, rather than using sets of points as its foundation, builds up topological concepts through the study of lattices, and, in particular, thecategory-theoretic study offrames and locales .**See also***

Glossary of general topology for detailed definitions

*List of general topology topics for related articles

*Category of topological spaces **References**Some standard books on general topology include:

*Bourbaki ; Topologie Générale (General Topology); ISBN 0-387-19374-X

*John L. Kelley ; General Topology; ISBN 0-387-90125-6

*James Munkres ; Topology; ISBN 0-13-181629-2

*Ryszard Engelking ; General Topology; ISBN 3-88538-006-4

* | year=1995

* O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev; Textbook in Problems on Elementary Topology; [*http://www.math.uu.se/~oleg/topoman.html online version*]The

arXiv subject code is [*http://arxiv.org/list/math.GN/recent math.GN*] .

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