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:
“Good resolutions are useless attempts to interfere with scientific laws. Their origin is pure vanity. Their result is absolutely nil. They give us, now and then, some of those luxurious sterile emotions that have a certain charm for the weak.... They are simply cheques that men draw on a bank where they have no account.”
—Oscar Wilde (1854–1900)
“Painting gives the object itself; poetry what it implies. Painting embodies what a thing contains in itself; poetry suggests what exists out of it, in any manner connected with it.”
—William Hazlitt (1778–1830)
“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)