= Wall metric
{title2=$d_{\mathcal W}$}
For a <wallspace> $(S,\mathcal W)$, define
$$
d_{\mathcal W}(x,y)=\#\{W\in\mathcal W:W\text{ separates }x\text{ and }y\}.
$$
The wallspace axiom makes this finite. It is always a <pseudometric>: it is symmetric, vanishes on the diagonal and satisfies the <triangle inequality>. For the last claim, every wall separating $x$ and $z$ must separate $x$ from $y$ or $y$ from $z$, so counting walls gives the inequality.<From wall spaces to CAT(0) cube complexes by Indira Chatterji and Graham Niblo>[Section 1, page 3]{ref}
It is a <metric> exactly when the walls separate every pair of distinct points. For example, on $S=\{a,b,c\}$ with only the wall $\{\{a,b\},\{c\}\}$, the distinct points $a,b$ have distance zero. Identifying pairs at distance zero gives a genuine <metric space>.
The <principal vertex of a dual cube complex> sends indistinguishable points to the same vertex. The wall-count distance equals the combinatorial distance between their principal vertices, making this <pseudometric> useful for the <wall-metric properness criterion>.
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