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:
“Religion means goal and way, politics implies end and means. The political end is recognizable by the fact that it may be attained—in success—and its attainment is historically recorded. The religious goal remains, even in man’s highest experiences, that which simply provides direction on the mortal way; it never enters into historical consummation.”
—Martin Buber (1878–1965)
“A peasant becomes fond of his pig and is glad to salt away its pork. What is significant, and is so difficult for the urban stranger to understand, is that the two statements are connected by an and and not by a but.”
—John Berger (b. 1926)
“The poetic notion of infinity is far greater than that which is sponsored by any creed.”
—Joseph Brodsky (b. 1940)