Monotone versions of some selection principles (Q1738936)

Space $X$ is \textsf{ SS-mM} if there is an operator assigning to every sequence $\mathcal U=\langle \mathcal U_n\rangle$ of open covers of $X$ a sequence $r(\mathcal U)=\langle r(\mathcal U)_n\rangle$ so that $\cup_nr(\mathcal U)_n$ is an open cover of $X$, $r(\mathcal U)_n$ refines $\mathcal U_n$ for every $n$ and for any two sequences $\mathcal U=\langle \mathcal U_n\rangle$ and $\mathcal V=\langle \mathcal V_n\rangle$ of open covers if $\mathcal U_n$ refines $\mathcal V_n$ for each $n$ then $r(\mathcal U)_n$ refines $r(\mathcal V)_n$ for each $n$. \textsf{ W} versions (replacing one or both \textsf{ S}) weaken refinement for individual members of the sequences to refinement for the union of members of the sequences. \textsf{ R} and \textsf{ H} (replacing \textsf{ M}) versions relate to the Hurewicz and Rothberger properties. Connections amongst these properties, as well as monotone Lindelöfness, are considered. It is also observed that the \textsf{ WS} versions are uninteresting and the \textsf{ WW} slightly less so. Local versions are also considered.