The map $\PSH(X,\theta)\to\mathbf{R}$ defined by $\varphi\mapsto\sup_{X}\varphi$ is continuous and proper. Hence $\PSH(X,\theta)/\mathbf{R}$ is compact. Furthermore, the topology on $\PSH(X,\theta)$ is equivalent to the topology of pointwise convergence on either $X^{\mathrm{qm}}$ or $X^{\mathrm{div}}$.