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The nerve lemma identifies a space with the geometric realization of the nerve of a sufficiently well-behaved good cover, up to homotopy equivalence. A standard version assumes a paracompact Hausdorff space with an open good cover: every nonempty finite intersection is contractible. The nerve has one vertex per covering set and one simplex per nonempty finite intersection.^[ref]

Cover a circle by three open arcs chosen so each pair overlaps in an interval and there is no triple intersection. The nerve has three vertices and three edges, but no filled triangle, so it has the homotopy type of a circle. Filling in that triangle would incorrectly make it contractible. The example shows why one must track all intersections, rather than only whether the individual covering sets are contractible.

For a connected $n$-dimensional manifold without boundary,Choose a locally finite good cover by coordinate disks, each with a local orientation. Compactly supported classes involve only finitely many of these disks at a time. The compact-support Mayer–Vietoris sequence gives one generator $u_i$ per disk and overlap relations $u_i=\pm u_j$, with the sign determined by the two local orientations. Passing to the direct limit over finite unions handles a noncompact manifold; a finite good cover of the whole manifold need not exist.

The overlap graph is connected, so all generators reduce to one up to sign. If local orientations can be chosen coherently, every relation has positive sign and the result is $\mathbb{Z}$. An orientation-reversing loop imposes $u=-u$, giving $\mathbb{Z}/2$. This is also the top-degree case of Poincare duality with the orientation local system, using compactly supported cohomology on the cohomology side.