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:
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—William Howard Taft (1857–1930)
“Happy is that mother whose ability to help her children continues on from babyhood and manhood into maturity. Blessed is the son who need not leave his mother at the threshold of the world’s activities, but may always and everywhere have her blessing and her help. Thrice blessed are the son and the mother between whom there exists an association not only physical and affectional, but spiritual and intellectual, and broad and wise as is the scope of each being.”
—Lydia Hoyt Farmer (1842–1903)
“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)