General topology

In mathematics, general topology or point-set topology is the branch of topology which 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 provides the most general framework where fundamental concepts of topology such as open/closed sets, continuity, interior/exterior/boundary points, and limit points could be defined.

Definition

Main article: Topological Space

A topology is a pair (X,Σ) consisting of a set X and a collection Σ of subsets of X, called open sets, satisfying the following three axioms:

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. X and the empty set ∅ are open sets.

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 the manifold concept
• the study of metric spaces, especially 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.

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 sets;
• interior and closure;
• neighbourhood and closeness;
• compactness and connectedness;
• continuous functions;
• convergence of sequences, nets, and filters;
• separation axioms
• 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 topology examines 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-theoretic study of frames and locales.

See also

• List of examples in general topology
• 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
• Paul L. Shick; Topology: Point-Set and Geometric; ISBN 0-470-09605-5
• Ryszard Engelking; General Topology; ISBN 3-88538-006-4
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR507446 
• O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev; Elementary Topology: Textbook in Problems; ISBN 978-0-8218-4506-6

The arXiv subject code is math.GN.