The Salvetti order complex and its finite CW case

Put $V=P(\mathcal{A})$ and let $K$ consist of the empty set and its nonempty finite chains. Every subset of a finite chain is a finite chain or empty, and every vertex is a singleton chain. Thus $(V,K)$ meets the abstract simplicial complex definition. Write $X=\mathrm{\mid }K\mathrm{\mid }$ for its weak realization.

For each vertex $v$, the coordinate function $a_v:X\to [0,1]$, $a_v(\alpha )=\alpha (v)$, is continuous: its restriction to every Euclidean simplex is continuous, so inverse images of open sets are open by the weak topology. Distinct points differ at some coordinate; inverse images of disjoint open intervals about those coordinate values are disjoint open neighbourhoods. Thus $X$ is Hausdorff. Simplex inclusions are continuous by the same weak-topology definition, and their inverse barycentric coordinate maps are continuous by the coordinate functions. Hence every simplex has its Euclidean topology as a subspace of $X$.

For a nonempty chain $\sigma$ of size $p+1$, the points of support exactly $\sigma$ form its open simplex $e_{\sigma }$. These sets partition $X$. The set $\mathrm{\mid }\sigma \mathrm{\mid }$ is closed in $X$: it is the intersection of the coordinate zero sets for vertices outside $\sigma$. Since $e_{\sigma }$ is dense in its Euclidean simplex, its closure in $X$ is exactly $\mathrm{\mid }\sigma \mathrm{\mid }$. The standard $p$-simplex is a closed $p$-disk with boundary the union of its proper faces, giving the required disk characteristic map and an interior homeomorphism onto $e_{\sigma }$.

Define $X^{-1}=\mathrm{\varnothing }$ and let $X^p$ be the union of the simplices of dimension at most $p$. These form an increasing filtration with union $X$, since every point has finite support. The subspace topology on $X^p$ is the weak topology on those simplices. Indeed, if $B\subseteq X^p$ has closed intersection with each of them, then for any simplex $\tau$ of $K$, $B\cap \mathrm{\mid }\tau \mathrm{\mid }$ is the finite union of the closed sets $B\cap \mathrm{\mid }\sigma \mathrm{\mid }$ over the faces $\sigma \subseteq \tau$ of dimension at most $p$. Each such face is closed in $\mathrm{\mid }\tau \mathrm{\mid }$, so the weak topology makes $B$ closed in $X$, hence in $X^p$. Conversely, a set closed in $X^p$ has closed intersection with each of its simplices by continuity of inclusion. For $p=0$, every subset meets each vertex simplex in a closed set, so every subset of $X^0$ is closed and $X^0$ is discrete.

For $p\ge 1$, take the disjoint union of $X^{p-1}$ and a closed $p$-simplex for each $p$-dimensional simplex of $K$, and identify every boundary point with the same barycentric point in $X^{p-1}$. Face inclusions give continuous attaching maps. The resulting map onto $X^p$ identifies exactly these points, since distinct $p$-simplex interiors have distinct supports. A subset of $X^p$ has closed preimage in this disjoint union exactly when its intersection with $X^{p-1}$ and with each $p$-simplex is closed. By the preceding skeleton criterion, this is exactly closedness in $X^p$. The map therefore has the quotient topology, proving that each skeleton is obtained by the disk attachments required in [F5].

The closure $\mathrm{\mid }\sigma \mathrm{\mid }$ of each cell meets precisely its finitely many nonempty face cells. This proves closure finiteness. Closedness in $X$ is tested on all the Euclidean simplices by [F2]; these are exactly the closed cells with their subspace topology. This proves the CW weak-topology condition. Together with Hausdorffness, the discrete zero-skeleton and the disk attachments, all conditions in [F3] hold.

If $\mathrm{\mid }\mathcal{A}\mathrm{\mid }=n$, distinct facets have distinct sign vectors in $\{-,0,+{\}}^{\mathcal{A}}$, and distinct chambers have distinct sign vectors in $\{-,+{\}}^{\mathcal{A}}$, by their fiber definitions. Thus there are at most $3^n$ facets and $2^n$ chambers, and at most $6^n$ incident pairs. A finite vertex set has finitely many subsets, hence finitely many chains and cells. This also includes $n=0$, since the sign map then has the single nonempty fiber $I$, which is also its single chamber.