In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for all compact subsets C of X, there is a compact set D in X containing C so that the induced map
is trivial. Intuitively, this is the property that loops far away from a small subspace of X can be collapsed, no matter how bad the small subspace is.
The Whitehead manifold is an example of a 3-manifold that is contractible but not simply connected at infinity. Since this property is invariant under homeomorphism, this proves that the Whitehead manifold is not homeomorphic to R3. However, it is a theorem that any contractible n-manifold which is also simply connected at infinity is homeomorphic to Rn.
Famous quotes containing the words simply, connected and/or infinity:
“I simply contend that the middle-class ideal which demands that people be affectionate, respectable, honest and content, that they avoid excitements and cultivate serenity is the ideal that appeals to me, it is in short the ideal of affectionate family life, of honorable business methods.”
—Gertrude Stein (1874–1946)
“War and culture, those are the two poles of Europe, her heaven and hell, her glory and shame, and they cannot be separated from one another. When one comes to an end, the other will end also and one cannot end without the other. The fact that no war has broken out in Europe for fifty years is connected in some mysterious way with the fact that for fifty years no new Picasso has appeared either.”
—Milan Kundera (b. 1929)
“As we begin to comprehend that the earth itself is a kind of manned spaceship hurtling through the infinity of space—it will seem increasingly absurd that we have not better organized the life of the human family.”
—Hubert H. Humphrey (1911–1978)