Topological pair

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In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces i:AX. Sometimes i is assumed to be a cofibration. A morphism from (X,A) to (X,A) is given by two maps f:XX and g:AA such that ig=fi. A pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace. 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,[1] homology theory and cohomology theory, where chains in A are made equivalent to 0, when considered as chains in X. Heuristically, one often thinks of a pair (X,A) as being akin to the quotient space X/A. There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space X to the pair (X,). A related concept is that of a triple (X, A, B), with BAX. 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 x0BAX.[1]

References

  1. 1.0 1.1 Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.
  • Patty, C. Wayne (2009), Foundations of Topology (2nd ed.), p. 276.