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 spent eight years trying to reach him and then another seven trying to keep him locked up because I realized that what was lying behind that boy’s eyes was purely and simply evil.”
—John Carpenter (b. 1948)
“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)
“The poetic notion of infinity is far greater than that which is sponsored by any creed.”
—Joseph Brodsky (b. 1940)