Wall metric ($d_{\mathcal{W}}$)

For a wallspace $(S,\mathcal{W})$, defineThe 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.^[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.