In mathematics, more specifically

algebraic topology 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 classif ...

, a pair

(X,A)

is shorthand for an inclusion of

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

i\colon A \hookrightarrow X

. Sometimes

i

is assumed to be a cofibration. A morphism from

(X,A)

(X',A')

is given by two maps

f\colon X\rightarrow X'

and

g\colon A \rightarrow A'

such that

i' \circ g =f \circ i

. A pair of spaces is an ordered pair where is a topological space and a subspace (with the

subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...

). The use of pairs of spaces is sometimes more convenient and technically superior to taking a

quotient space Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...

of by . Pairs of spaces occur centrally in relative homology,

homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...

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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

from the

category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...

to the category of pairs of spaces, which sends a space

X

to the pair

(X, \varnothing)

. A related concept is that of a triple , with . Triples are used in

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...

. Often, for a pointed space with basepoint at , one writes the triple as , where .

References

*. Algebraic topology {{topology-stub