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:
“Only the more uncompromising of the mystics still seek for knowledge in a silent land of absolute intuition, where the intellect finally lays down its conceptual tools, and rests from its pragmatic labors, while its works do not follow it, but are simply forgotten, and are as if they never had been.”
—Josiah Royce (1855–1916)
“As long as learning is connected with earning, as long as certain jobs can only be reached through exams, so long must we take this examination system seriously. If another ladder to employment was contrived, much so-called education would disappear, and no one would be a penny the stupider.”
—E.M. (Edward Morgan)
“The poetic notion of infinity is far greater than that which is sponsored by any creed.”
—Joseph Brodsky (b. 1940)