
Topological pair
Encyclopedia
In mathematics
, more specifically algebraic topology
, a pair
is short-hand for an inclusion of topological spaces
. Sometimes
is assumed to be a cofibration
. A morphism from
to
is given by two maps
and
such that
.
Pairs come up mainly in 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
.
There is a functor from spaces to pairs, which sends a space
to the pair
.
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...
, more specifically 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



Cofibration
In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...
. A morphism from





Pairs come up mainly in homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
and cohomology theory, where chains in


Heuristically, one often thinks of a pair


There is a functor from spaces to pairs, which sends a space

