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:
“Amor fati: that shall henceforth be my love! I do not want to wage war against the ugly. I do not want to accuse—I do not even want to accuse those who accuse. Let looking away be my only denial! And all in all and on the whole: I want someday to be purely and simply a Yes-sayer!”
—Friedrich Nietzsche (1844–1900)
“The question of armaments, whether on land or sea, is the most immediately and intensely practical question connected with the future fortunes of nations and of mankind.”
—Woodrow Wilson (1856–1924)
“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)