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A good cover is an open cover in which every nonempty finite intersection of covering sets is contractible. In particular, each individual covering set must be contractible. Requiring only contractible sets, without their intersections, is weaker.

For example, two long arcs can cover a circle while their overlap consists of two separated arcs; this is not a good cover. Three suitably chosen shorter arcs cover the circle with pairwise interval overlaps and no triple overlap. That is a good cover whose nerve is a triangle boundary.

The nerve records a simplex for each nonempty finite intersection. For a smooth manifold with an open good cover, the nerve lemma identifies the homotopy type of the nerve with that of the manifold. Contractible intersections therefore permit local data to encode global topology combinatorially. Empty intersections contribute no simplex and impose no contractibility requirement.

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.