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:
“Upscale people are fixated with food simply because they are now able to eat so much of it without getting fat, and the reason they don’t get fat is that they maintain a profligate level of calorie expenditure. The very same people whose evenings begin with melted goat’s cheese ... get up at dawn to run, break for a mid-morning aerobics class, and watch the evening news while racing on a stationary bicycle.”
—Barbara Ehrenreich (b. 1941)
“When, in the course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume the powers of the earth, the separate and equal station to which the laws of nature and of nature’s God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.”
—Thomas Jefferson (1743–1826)
“The poetic notion of infinity is far greater than that which is sponsored by any creed.”
—Joseph Brodsky (b. 1940)