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A topological algebra over a topological field is a topological vector space together with a bilinear multiplication :, (,) that turns into an algebra ...
Classic applications of algebraic topology include: The Brouwer fixed point theorem : every continuous map from the unit n -disk to itself has a fixed point. The free rank of the n th homology group of a simplicial complex is the n th Betti number , which allows one to calculate the Euler–Poincaré characteristic .
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 ...
Algebra & Number Theory; Algebra Colloquium; Algebra i Logika; Algebra Universalis; Algebraic & Geometric Topology; Algebraic Combinatorics; American Journal of Mathematics; American Mathematical Monthly; Analysis and Applications; The Analyst, or, Mathematical Museum; Annales Academiae Scientiarum Fennicae. Mathematica; Annales de Gergonne ...
The real numbers form a topological group under addition. In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.
The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this programmatic idea is the étale topos of a scheme. Another illustration of ...
This corresponds also to the period where homological algebra and category theory were introduced for the study of topological spaces, and largely supplanted combinatorial methods. More recently the term combinatorial topology has been revived for investigations carried out by treating topological objects as composed of pieces as in the older ...
Similarly, a fibre bundle with discrete fibre is a local system, because each path lifts uniquely to a given lift of its basepoint. (The definition adjusts to include set-valued local systems in the obvious way). A local system of k-vector spaces on X is equivalent to a k-linear representation of (,).
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