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 don’t believe in failure. In itself, it doesn’t exist. We create it. We make ourselves fail.”
—Alice Foote MacDougall (1867–1945)
“Religious fervor makes the devil a very real personage, and anything awe-inspiring or not easily understood is usually connected with him. Perhaps this explains why, not only in the Ozarks but all over the State, his name crops up so frequently.”
—Administration in the State of Miss, U.S. public relief program (1935-1943)
“The poetic notion of infinity is far greater than that which is sponsored by any creed.”
—Joseph Brodsky (b. 1940)