General Topology - Scope

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.