= Nerve lemma
{title2=$X\simeq|N(\mathcal U)|$}
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.<Algebraic Topology by Allen Hatcher>[Corollary 4G.3, nerve of a cover]{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.
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