{{Short description|Concept in algebraic topology}} In mathematics, more specifically algebraic topology, a pair <math>(X,A)</math> is shorthand for an inclusion of topological spaces <math>i\colon A \hookrightarrow X</math>. Sometimes <math>i</math> is assumed to be a cofibration. A morphism from <math>(X,A)</math> to <math>(X',A')</math> is given by two maps <math>f\colon X\rightarrow X'</math> and <math>g\colon A \rightarrow A'</math> such that <math> i' \circ g =f \circ i </math>.

A '''pair of spaces''' is an ordered pair {{math|(''X'', ''A'')}} where {{math|''X''}} is a topological space and {{math|''A''}} a subspace. The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of {{math|''X''}} by {{math|''A''}}. Pairs of spaces occur centrally in relative homology,<ref name="hatcher">{{cite book | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | url = https://pi.math.cornell.edu/~hatcher/AT/ATpage.html}}</ref> homology theory and cohomology theory, where chains in <math>A</math> are made equivalent to 0, when considered as chains in <math>X</math>.

Heuristically, one often thinks of a pair <math>(X,A)</math> as being akin to the quotient space <math>X/A</math>.

There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space <math>X</math> to the pair <math>(X, \varnothing)</math>.

A related concept is that of a triple {{math|(''X'', ''A'', ''B'')}}, with {{math|''B'' ⊂ ''A'' ⊂ ''X''}}. Triples are used in homotopy theory. Often, for a pointed space with basepoint at {{math|''x''<sub>0</sub>}}, one writes the triple as {{math|(''X'', ''A'', ''B'', ''x''<sub>0</sub>)}}, where {{math|''x''<sub>0</sub> ∈ ''B'' ⊂ ''A'' ⊂ ''X''}}.<ref name="hatcher" />

==References== {{Reflist}} *{{citation|title=Foundations of Topology|edition=2nd|first=C. Wayne|last=Patty|year=2009|page=276}}.

Category:Algebraic topology

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