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:
“Emperor Joseph II: Your work is ingenious. It’s quality work, and there are simply too many notes, that’s all. Just cut a few and it will be perfect.
Mozart: Which few did you have in mind, majesty?”
—Peter Shaffer (b. 1926)
“Nothing fortuitous happens in a child’s world. There are no accidents. Everything is connected with everything else and everything can be explained by everything else.... For a young child everything that happens is a necessity.”
—John Berger (b. 1926)
“The poetic notion of infinity is far greater than that which is sponsored by any creed.”
—Joseph Brodsky (b. 1940)