mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, 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 points ...

i\colon A \hookrightarrow X

. Sometimes

i

is assumed to be a

cofibration In mathematics, in particular homotopy theory, a continuous mapping :i: A \to X, where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,S/math> be extended to homotopy classes of maps ,S/math> whenever a map ...

. 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 to ...

). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of by . Pairs of spaces occur centrally in

relative homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...

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 topolog ...

and

cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

, 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 Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

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 contin ...

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 topolog ...

. Often, for a

pointed space In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...

with basepoint at , one writes the triple as , where .

References

*. Algebraic topology {{topology-stub