= Good cover
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.
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