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:
“The majority of persons choose their wives with as little prudence as they eat. They see a trull with nothing else to recommend her but a pair of thighs and choice hunkers, and so smart to void their seed that they marry her at once. They imagine they can live in marvelous contentment with handsome feet and ambrosial buttocks. Most men are accredited fools shortly after they leave the womb.”
—Edward Dahlberg (1900–1977)
“So people and things don’t pair anymore
With what they used to pair with before.”
—Robert Frost (1874–1963)
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)