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:
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—Oliver Stone (b. 1946)
“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)
“We must not suppose that, because a man is a rational animal, he will, therefore, always act rationally; or, because he has such or such a predominant passion, that he will act invariably and consequentially in pursuit of it. No, we are complicated machines; and though we have one main spring that gives motion to the whole, we have an infinity of little wheels, which, in their turns, retard, precipitate, and sometime stop that motion.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)