Algebraic topology

Topological pair

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 . A pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace (with the subspace topology). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of X by A. Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains in are made equivalent to 0, when considered as chains in . Heuristically, one often thinks of a pair as being akin to the quotient space . There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space to the pair . A related concept is that of a triple (X, A, B), with B ⊂ A ⊂ X. Triples are used in homotopy theory. Often, for a pointed space with basepoint at x0, one writes the triple as (X, A, B, x0), where x0 ∈ B ⊂ A ⊂ X. (Wikipedia).

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Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of a Topological Space

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From playlist Basics: Topology

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From playlist Topology

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From playlist Parallel Lines and a Transversal

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From playlist Basics: Topology

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From playlist Global Noncommutative Geometry Seminar (Europe)

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From playlist Topology

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From playlist Parallel Lines and a Transversal

Related pages

Functor | Quotient space (topology) | Topological space | Category of topological spaces | Mathematics | Subspace topology | Relative homology | Homotopy theory | Cofibration | Pointed space | Algebraic topology