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)
“Auld Noah was at hame wi’ them a’,
The lion and the lamb,
Pair by pair they entered the Ark
And he took them as they cam’.”
—Hugh MacDiarmid (1892–1978)
“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)