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A topological algebra over a topological field is a topological vector space together with a bilinear multiplication ⋅ : A × A → A {\displaystyle \cdot :A\times A\to A} , ( a , b ) ↦ a ⋅ b {\displaystyle (a,b)\mapsto a\cdot b}
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. Although algebraic topology primarily uses algebra to study topological ...
Path (topology) Fundamental group; Homotopy group; Seifert–van Kampen theorem; Pointed space; Winding number; Simply connected. Universal cover; Monodromy; Homotopy lifting property; Mapping cylinder; Mapping cone (topology) Wedge sum; Smash product; Adjunction space; Cohomotopy; Cohomotopy group; Brown's representability theorem; Eilenberg ...
The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. This is a list of topology topics. See also: Topology glossary; List of topologies; List of general topology topics; List of geometric topology topics
For example, a sphere has two cells: one 0-cell and one -cell, since can be obtained by collapsing the boundary of the n-disk to a point. In general, every manifold has the homotopy type of a CW complex; [ 3 ] in fact, Morse theory implies that a compact manifold has the homotopy type of a finite CW complex.
This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology ), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are ...
More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. The basic example of a topos comes from the Zariski topos of a scheme . For each scheme X {\displaystyle X} there is a site Open ( X ) {\displaystyle {\text{Open}}(X)} (of objects given by open subsets and morphisms given by inclusions) whose category of ...
In mathematics, more specifically algebraic topology, a pair (,) is shorthand for an inclusion of topological spaces:.Sometimes is assumed to be a cofibration.A morphism from (,) to (′, ′) is given by two maps : ′ and : ′ such that ′ =.
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