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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 ...
In homotopy theory and algebraic topology, the word "space" denotes a topological space.In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.
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
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 the above example, a connection with classical Galois theory can be seen by regarding ^ as the profinite Galois group Gal(F /F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F .
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 ...
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 ′ =.