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:
“Oh, Jacques, we’re used to each other, we’re a pair of captive hawks caught in the same cage, and so we’ve grown used to each other. That’s what passes for love at this dim, shadowy end of the Camino Real.”
—Tennessee Williams (1914–1983)
“Oh, Jacques, we’re used to each other, we’re a pair of captive hawks caught in the same cage, and so we’ve grown used to each other. That’s what passes for love at this dim, shadowy end of the Camino Real.”
—Tennessee Williams (1914–1983)
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)