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 Republican convention, an event with the intellectual content of a Guns’n’Roses lyric attended by every ofay insurance broker in America who owns a pair of white shoes.”
—P.J. (Patrick Jake)
“I saw a guide-post surmounted by a pair of moose horns.... They are sometimes used for ornamental hat-trees, together with deer’s horns, in front entries; but ... I trust that I shall have a better excuse for killing a moose than that I may hang my hat on his horns.”
—Henry David Thoreau (1817–1862)
“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)