  The complex Monge-Amp\`ere operator has been defined for locally bounded
plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has
been extended to compact complex manifolds, and to various classes of mildly
unbounded quasi-plurisubharmonic functions by various authors. As this operator
is not continuous for the $L^{1}$-topology, several stronger topologies have
been introduced over the last decades to remedy this, while maintaining
efficient compactness criteria. The purpose of this note is to show that these
stronger topologies are essentially equivalent to the natural quasi-monotone
topology that we introduce and study here.
