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:
“Not the less does nature continue to fill the heart of youth with suggestions of his enthusiasm, and there are now men,—if indeed I can speak in the plural number,—more exactly, I will say, I have just been conversing with one man, to whom no weight of adverse experience will make it for a moment appear impossible, that thousands of human beings might exercise towards each other the grandest and simplest of sentiments, as well as a knot of friends, or a pair of lovers.”
—Ralph Waldo Emerson (1803–1882)
“What a pair of spectacles is here!”
—William Shakespeare (1564–1616)
“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)