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:
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.
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)
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces.
Topology is concerned with the study of the so-called topological properties of figures, that is to say, properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms).
In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
In mathematics, generaltopology or point-set topology is the branch of topology which studies elementary properties of topological spaces and structures defined on them.
Set-theoretic topology examines such questions when they have substantial relations to axiomatic set theory, as is often the case.
Other main branches of topology are algebraic topology, geometric topology, and differential topology.
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