Nerve of a cover

Remarks

If $J$ is a face and $J' \subseteq J$ is nonempty, then $\bigcap_{j \in J'} U_j \supseteq \bigcap_{j \in J} U_j \ne \emptyset$, so $J'$ is a face; this is why $N(\mathcal{U})$ is a simplicial complex.

Example: cover the circle $S^1$ by three open arcs $U_0$, $U_1$, $U_2$, each a bit longer than one third of the circle. Choose them so that consecutive arcs overlap and no point lies in all three. Each pair $\set{i, j}$ is a face, and $\set{0, 1, 2}$ is not, so $N(\mathcal{U})$ is the boundary of a triangle.

Non-example: the complex with vertices $0, 1, 2$ and all subsets as faces is not the nerve of this cover, since it contains $\set{0, 1, 2}$ although $U_0 \cap U_1 \cap U_2 = \emptyset$.

The same definition applies to any family of sets; the nerve of a closed cover or of a family of convex sets is often used in combinatorics.