---
tag: P2F66BS71T
kind: definition
name: Nerve of a cover
msc: [55U10, 55N05]
defines: 4P3S0DRZXB
uses: [V8GZ0W1ZZ5, 049PXYHW1T, 0CNQXVBHCX]
sources:
  - url: https://en.wikipedia.org/wiki/Nerve_complex
---

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $\mathcal{U} = (U_i)_{i \in I}$ be a family of subsets of $X$, usually an [[open cover|049PXYHW1T]].
The *nerve* $N(\mathcal{U})$ is the abstract [[simplicial complex|V8GZ0W1ZZ5]] with vertex set $\set{i \in I : U_i \ne \emptyset}$ whose faces are the finite nonempty subsets $J \subseteq I$ such that
$$\bigcap_{j \in J} U_j \ne \emptyset.$$

## 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.