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:
“Goddesses never die. They slip in and out of the world’s cities, in and out of our dreams, century after century, answering to different names, dressed differently, perhaps even disguised, perhaps idle and unemployed, their official altars abandoned, their temples feared or simply forgotten.”
—Phyllis Chesler (b. 1941)
“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)
“New York, you are an Egypt! But an Egypt turned inside out. For she erected pyramids of slavery to death, and you erect pyramids of democracy with the vertical organ-pipes of your skyscrapers all meeting at the point of infinity of liberty!”
—Salvador Dali (1904–1989)