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:
“The long discussions and painful arguments of adolescence and the fierce loyalties to teachers, heroes, and gurus during the teenage years are simply our children’s struggles to ensure that the lifestyles and values they adopt are worthy of their allegiance.”
—Neil Kurshan (20th century)
“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)
“As we begin to comprehend that the earth itself is a kind of manned spaceship hurtling through the infinity of space—it will seem increasingly absurd that we have not better organized the life of the human family.”
—Hubert H. Humphrey (1911–1978)