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:
“A country survives its legislation. That truth should not comfort the conservative nor depress the radical. For it means that public policy can enlarge its scope and increase its audacity, can try big experiments without trembling too much over the result. This nation could enter upon the most radical experiments and could afford to fail in them.”
—Walter Lippmann (1889–1974)
“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)
“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)