Chambers, walls and adjacency

PDF p. 4: §2.1; PDF p. 9: Remark 3.8

Let $(T,\mathcal{H})$ be a hyperplane arrangement in a finite-dimensional real vector space $E$ with its Euclidean topology, as in Linear hyperplane arrangements and local finiteness and Locally finite arrangements and relative face terminology. Thus $T$ is convex and each member of $\mathcal{H}$ is a linear hyperplane meeting $T$. Give the complement $T\setminus {\bigcup }_{H\in \mathcal{H}}H$ its subspace topology. Its connected components are the chambers of the arrangement.

For a chamber $C$, let $\bar{C}$ be its closure in $E$: the set of points every open neighbourhood of which meets $C$. The affine hull of a nonempty set is its smallest containing affine subspace, and its affine dimension is the dimension of that hull. A wall of $C$ is an intersection $\bar{C}\cap H$, for $H\in \mathcal{H}$, whose affine dimension is one less than the affine dimension of $C$. Two distinct chambers are adjacent if they share a wall. The wall is the intersection itself; the hyperplane $H$ is its supporting hyperplane and is not the wall by this convention.

The arrangement definition supplies the convex domain and its hyperplanes; the subspace topology and connected components specify the chambers. Closure and affine dimension specify which intersections are walls without asserting that every hyperplane supplies one. If $C$ has affine dimension $\dim E$, this wall condition agrees with the dimension-$\dim E-1$ convention in the arrangement interface. Sharing one of these sets is a well-defined symmetric relation on distinct chambers.

The cited article supplies the chamber, wall and adjacency terminology. The ambient closure and affine-dimension convention make the formulation explicit; in the relative arrangement of §3.1 the ambient space is the relative vector space ${\mathrm{\Theta }}_J$.