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:
“And yet we constantly reclaim some part of that primal spontaneity through the youngest among us, not only through their sorrow and anger but simply through everyday discoveries, life unwrapped. To see a child touch the piano keys for the first time, to watch a small body slice through the surface of the water in a clean dive, is to experience the shock, not of the new, but of the familiar revisited as though it were strange and wonderful.”
—Anna Quindlen (b. 1952)
“Painting gives the object itself; poetry what it implies. Painting embodies what a thing contains in itself; poetry suggests what exists out of it, in any manner connected with it.”
—William Hazlitt (1778–1830)
“We must not suppose that, because a man is a rational animal, he will, therefore, always act rationally; or, because he has such or such a predominant passion, that he will act invariably and consequentially in pursuit of it. No, we are complicated machines; and though we have one main spring that gives motion to the whole, we have an infinity of little wheels, which, in their turns, retard, precipitate, and sometime stop that motion.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)