$\Sigma_s$-products revisited

summary:We show that any $\Sigma_s$-product of at most $\mathfrak{c}$-many $L\Sigma(\leq \omega)$-spaces has the $L\Sigma(\leq \omega)$-property. This result generalizes some known results about $L\Sigma(\leq \omega)$-spaces. On the other hand, we prove that every $\Sigma_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every $\Sigma_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [{Lifting the Collins-Roscoe property by condensations}, Topology Proc. {42} (2012), 1--15]. Besides, we prove that if $X$ is a simple Lindelöf $\Sigma$-space, then $C_p(X)$ has the Collins-Roscoe property