Simply connected at infinity

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Let X be 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

$\pi_1(X-D) \to \pi_1(X-C)$

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 which 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 R^3. However, it is a theorem that any contractible n-manifold which is also simply connected at infinity is homeomorphic to R^n.