Lemma 1

The set of hyperplanes $\mathscr{H}$ is locally finite.

Indeed, let K be a compact subset of E. If a hyperplane H ∈ $\mathscr{H}$ meets K, the set s_H(K) also meets K, since every point of K ∩ H is fixed by s_H. The set of H ∈ $\mathscr{H}$ meeting K is thus finite by (D'2).

We can thus apply to E and $\mathscr{H}$ the definitions and results of $§ 1$. We shall call the chambers, facets, walls, etc. defined in E by $\mathscr{H}$ simply the chambers, facets, walls, etc. relative to W. Any displacement w ∈ W permutes the chambers, facets, walls, etc.