Nerve lemma ($X\simeq |N(\mathcal{U})|$)

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.