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:
“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)
“For it is not the bare words but the scope of the writer that gives the true light, by which any writing is to be interpreted; and they that insist upon single texts, without considering the main design, can derive no thing from them clearly.”
—Thomas Hobbes (1579–1688)
“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)