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:
“When it’s over I don’t want to wonder
if I have made of my life something particular, and real.
I don’t want to find myself sighing and frightened,
or full of argument.
I don’t want to end up simply having visited this world.”
—Mary Oliver (b. 1935)
“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)
“New York, you are an Egypt! But an Egypt turned inside out. For she erected pyramids of slavery to death, and you erect pyramids of democracy with the vertical organ-pipes of your skyscrapers all meeting at the point of infinity of liberty!”
—Salvador Dali (1904–1989)