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 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, esp.
normed linear spaces, in the early days of
functional analysisIt 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 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
closed sets;
*
interior and
closure;
*
neighbourhood and
closeness;
*
compactness and
connectedness;
*
continuous functions;
*
convergence of
sequences,
nets, and
filters;
*
separation axioms
*
countability axioms
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.
*
Glossary of general topology for detailed definitions
*
List of general topology topics for related articles
*
Category of topological spacesSome 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
*
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
