- General topology
mathematics, general topology or point-set topology is the branch of topologywhich studies properties of topological spaces 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 to manifolds.
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 the
*the study of
metric spaces, esp. normed linear spaces, in the early days of functional 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.
More specifically, it is in general topology that basic notions are defined and theorems about them proved. This includes the following:
* open and
* interior and closure;
* neighbourhood and closeness;
* compactness and connectedness;
* continuous functions;
* convergence of
sequences, nets, and filters;
* countability axiom
Other more advanced notions also appear, but are usually related directly to these fundamental concepts, without reference to other branches of mathematics.
Set-theoretic topologyexamines such questions when they have substantial relations to set theory, as is often the case.
Other main branches of topology are
algebraic topology, geometric topology, and differential 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, the category-theoreticstudy of frames and locales.
Glossary of general topologyfor detailed definitions
List of general topology topicsfor related articles
Category of topological spaces
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]
arXivsubject code is [http://arxiv.org/list/math.GN/recent math.GN] .
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