  Let $X$ be a compact K\"ahler manifold. Given a big cohomology class
$\{\theta\}$, there is a natural equivalence relation on the space of
$\theta$-psh functions giving rise to $\mathcal S(X,\theta)$, the space of
singularity types of potentials. We introduce a natural pseudometric
$d_{\mathcal {S}}$ on $\mathcal S(X,\theta)$ that is non-degenerate on the
space of model singularity types and whose atoms are exactly the relative full
mass classes. In the presence of positive mass we show that this metric space
is complete. As applications, we show that solutions to a family of complex
Monge-Amp\`ere equations with varying singularity type converge as governed by
the $d_\mathcal S$-topology, and we obtain a semicontinuity result for
multiplier ideal sheaves associated to singularity types, extending the scope
of previous results from the local context.
