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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 lattice of topologies on a set is a complemented lattice; that is, given a topology on there exists a topology ′ on such that the intersection ′ is the trivial topology and the topology generated by the union ′ is the discrete topology. [3] [4] If the set has at least three elements, the lattice of topologies on is not modular, [5 ...
A torus, one of the most frequently studied objects in algebraic topology. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
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 .
Manifolds have a rich set of invariants, including: Point-set topology. Compactness; Connectedness; Classic algebraic topology. Euler characteristic; Fundamental group; Cohomology ring; Geometric topology. normal invariants (orientability, characteristic classes, and characteristic numbers) Simple homotopy (Reidemeister torsion) Surgery theory
Cardinal functions are widely used in topology as a tool for describing various topological properties. [4] [5] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [6] prefer to define the cardinal functions listed below so that they never take on finite cardinal numbers as values; this requires modifying some of the definitions ...