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:
“... But all the feelings that evoke in us the joy or the misfortune of a real person are only produced in us through the intermediary of an image of that joy or that misfortune; the ingeniousness of the first novelist was in understanding that, in the apparatus of our emotions, since the image is the only essential element, the simplification which consists of purely and simply suppressing the factual characters is a definitive improvement.”
—Marcel Proust (1871–1922)
“A peasant becomes fond of his pig and is glad to salt away its pork. What is significant, and is so difficult for the urban stranger to understand, is that the two statements are connected by an and and not by a but.”
—John Berger (b. 1926)
“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)