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:
“One of the most horrible, yet most important, discoveries of our age has been that, if you really wish to destroy a person and turn him into an automaton, the surest method is not physical torture, in the strict sense, but simply to keep him awake, i.e., in an existential relation to life without intermission.”
—W.H. (Wystan Hugh)
“We can’t nourish our children if we don’t nourish ourselves.... Parents who manage to stay married, sane, and connected to each other share one basic characteristic: The ability to protect even small amounts of time together no matter what else is going on in their lives.”
—Ron Taffel (20th century)
“The poetic notion of infinity is far greater than that which is sponsored by any creed.”
—Joseph Brodsky (b. 1940)