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.
Read more about this topic: General Topology
Famous quotes containing the word scope:
“In the works of man, everything is as poor as its author; vision is confined, means are limited, scope is restricted, movements are labored, and results are humdrum.”
—Joseph De Maistre (1753–1821)
“As the creative adult needs to toy with ideas, the child, to form his ideas, needs toys—and plenty of leisure and scope to play with them as he likes, and not just the way adults think proper. This is why he must be given this freedom for his play to be successful and truly serve him well.”
—Bruno Bettelheim (20th century)
“Every person is responsible for all the good within the scope of his abilities, and for no more, and none can tell whose sphere is the largest.”
—Gail Hamilton (1833–1896)