[BFJ11, Theorem 7.8]. For each model function $\psi$ the map $\varphi\mapsto\sup_{X}(\varphi-\psi)$ is continuous and proper on $\PSH(X,\theta)$. In particular, the space $\PSH(X,\theta)/\mathbf{R}$ is compact. Further, the topology on $\PSH(X,\theta)$ is equivalent to the topology of pointwise convergence on $X^{\mathrm{div}}$.