In mathematics, particularly algebraic topology, a pair of spaces is an ordered pair (X, A) where X is a topological space and A 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 X by A. Pairs of spaces occur centrally in relative homology.
A related concept is that of a triple (X, A, B), with B ⊂ A ⊂ X. Triples are used in homotopy theory. Often, for a pointed space with basepoint at x0, one writes the triple as (X, A, B, x0), where x0 ∈ B ⊂ A ⊂ X.
Famous quotes containing the words pair of, pair and/or spaces:
“Here comes a pair of very strange beasts, which in all
tongues are called fools.”
—William Shakespeare (1564–1616)
“... the Ovarian Theory of Literature, or, rather, its complement, the Testicular Theory. A recent camp follower ... of this explicit theory is ... Norman Mailer, who has attributed his own gift, and the literary gift in general, solely and directly to the possession of a specific pair of organs. One writes with these organs, Mailer has said ... and I have always wondered with what shade of ink he manages to do it.”
—Cynthia Ozick (b. 1928)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)