= Disconnected at infinity
A connected noncompact <manifold> is disconnected at infinity if removing some compact <subset> leaves at least two connected components that are not relatively compact in the manifold. Equivalently, the manifold has at least two <End (topology)>[ends]: there are distinct ways to leave every compact <subset>.
The <real line> has two <End (topology)>[ends], while <Euclidean space> $\mathbb R^n$ for $n\ge2$ has one. The <circular cylinder> also has two, corresponding to the two axial directions. In a proper <metric space>, components are not relatively compact exactly when they are unbounded, so the familiar unbounded-component formulation applies to complete connected <Riemannian manifolds> by the <Hopf-Rinow theorem>. Relative compactness gives the intrinsic definition in <topology> without imposing a particular metric: an open <Euclidean ball> is bounded in its inherited <distance> but still has an end.
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