In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. It grew out of a number of areas, such as the detailed study of sets of points (as subsets of the real line, understood), the manifold concept, the metric spaces and the early days of functional analysis. It was codified, in much its form for the remainder of the twentieth century, around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in every area of mathematics.

More specifically, it is in general topology that basic notions, such as:

• open and closed sets;
• compact spaces;
• continuous functions;
• convergence of sequences, nets, and filters;
• separation axioms:

are defined and theorems about them are proved.

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

See glossary of general topology for detailed definitions; and the list of general topology topics.

Standard references

Some standard books on general topology include:

• Bourbaki; Topologie Générale (General Topology); ISBN 0-387-19374-X
• John Kelley; General Topology; ISBN 0-387-90125-6
• James Munkres; Topology; ISBN 0-13-181629-2
• Lynn Steen & Arthur Seebach; Counterexamples in Topology; ISBN 0-486-68735-X
• O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev; Textbook in Problems on Elementary Topology; online version (http://www.math.uu.se/~oleg/educ-texts.html)