each nonempty ${{\mathcal{J}}}\subset [n]$. Then $$\|{{\mathcal{N}}}(\{A_1,A_2,\dots,A_n\})\| \simeq \|K\|;$$ i.e., the nerve is homotopy equivalent to $K$.

Meet semi-lattices {#secmeet}
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*A *meet-semilattice* $(L,\wedge)$ is a set $L$ with a binary operation $\wedge\colon L\times L\to L$ satisfying*

1.  (associativity) $w\wedge (x \wedge y) = (w\wedge x)\wedge y$,

2.  (symmetry) $w\wedge x = x\wedge w$, and

3.  (idempotency) $w\wedge w = w$.

A classical example of a meet-semilattice comes from set theory. Given a set $X$, let $2^X$ denote its power set. Then, $(2^X,\cap)$ is a meet-semilattice.

Indexing sets {#secindexing}
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As indicated in the Introduction the proof of Theorem \[thm:main\] makes use of a complicated class of coverings. The remainder of this section introduces the notation used to index these coverings.

\[def:calL\]

*For $n\leq N$, we denote the set of [*multi-indices*]{} by $\mathbb{L}_{2n}^N\subset \{1,\dots,N\}^{2n}$ and define it to consist of the vectors $w=( w_1,\dots,w_{2n} )$ that satisfy the following four conditions for all $k=1,\ldots, n-1$.*

1.  $w_{2k-1}\leq w_{2k}$

2.  $w_{2k-1}< w_{2k+1}$

3.  $w_{2k} < w_{2k+2}$

4.  $w_{2k+1} \leq w_{2k+2}$.

The order relation in $\mathbb{L}_{2n}^N$ is summarized in Figure \[fig:Lorder\]. Condition (ii) implies that the assumption that $n\leq N$ is necessary. Observe that if $n=N$, then $\mathbb{L}_{2n}^N = \{(1,1,2,2,\ldots,N,N)\}$ consists of a 