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,[1]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 .
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.[1]
topological, pair, mathematics, more, specifically, algebraic, topology, pair, displaystyle, shorthand, inclusion, topological, spaces, displaystyle, colon, hookrightarrow, sometimes, displaystyle, assumed, cofibration, morphism, from, displaystyle, displaysty. In mathematics more specifically algebraic topology a pair X A displaystyle X A is shorthand for an inclusion of topological spaces i A X displaystyle i colon A hookrightarrow X Sometimes i displaystyle i is assumed to be a cofibration A morphism from X A displaystyle X A to X A displaystyle X A is given by two maps f X X displaystyle f colon X rightarrow X and g A A displaystyle g colon A rightarrow A such that i g f i displaystyle i circ g f circ i 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 1 homology theory and cohomology theory where chains in A displaystyle A are made equivalent to 0 when considered as chains in X displaystyle X Heuristically one often thinks of a pair X A displaystyle X A as being akin to the quotient space X A displaystyle X A There is a functor from the category of topological spaces to the category of pairs of spaces which sends a space X displaystyle X to the pair X displaystyle X varnothing 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 1 References edit a b Hatcher Allen 2002 Algebraic Topology Cambridge University Press ISBN 0 521 79540 0 Patty C Wayne 2009 Foundations of Topology 2nd ed p 276 nbsp This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Topological pair amp oldid 995427980, wikipedia, wiki, book, books, library,