Pair of spaces
Encyclopedia
In mathematics
, particularly algebraic topology
, a pair of spaces is an ordered pair where is a topological space
and 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 by . Pairs of spaces occur centrally in relative homology
.
A related concept is that of a triple , with . Triples are used in homotopy theory. Often, for a pointed space
with basepoint at , one writes the triple as , where .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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 classify up to homotopy equivalence.Although algebraic topology...
, a pair of spaces is an ordered pair where is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
and a subspace (with the subspace topology
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 natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
).
The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of by . Pairs of spaces occur centrally in relative homology
Relative homology
In algebraic topology, a branch of mathematics, the 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...
.
A related concept is that of a triple , with . Triples are used in homotopy theory. Often, for a pointed space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
with basepoint at , one writes the triple as , where .