# Fundamental group at infinity

From Topospaces

## Definition

### Loose 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.

Note that this loose definition is not quite precise, because the choice of basepoint *does* affect the choice of map, and if complements of compact sets are not path-connected, then we need to be careful about choosing the fundamental group.

## 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.