Fundamental group at infinity

Definition

The fundamental group at infinity of a path-connected space is the inverse limit of the fundamental groups of complements of compact subsets. For a compact space, the fundamental group at infinity is trivial.

Facts

The fundamental group at infinity is not homotopy-invariant. In fact, there exist contractible spaces whose fundamental group at infinity does not vanish. Thus, the fundamental group at infinity is a tool to distinguish between non-homeomorphic spaces which are homotopy-equivalent.