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In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology , geometric topology , and algebraic topology .
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
The following topologies are a known source of counterexamples for point-set topology. Alexandroff plank; Appert topology − A Hausdorff, perfectly normal (T 6), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact. Arens square
The definition of a point of closure of a set is closely related to the definition of a limit point of a set.The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to ...
In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if = ′, where ′ denotes the set of all limit points of , also known as the derived set of . (Some authors do not consider the empty set to be perfect.
A set in the plane is a neighbourhood of a point if a small disc around is contained in . The small disc around is an open set .. In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous.
Trivial topology; Cofinite topology; Finer topology; Product topology. Restricted product; Quotient space; Unit interval; Continuum (topology) Extended real number line; Long line (topology) Sierpinski space; Cantor set, Cantor space, Cantor cube; Space-filling curve; Topologist's sine curve; Uniform norm; Weak topology; Strong topology ...