  In this paper we investigate R-,H-, and M-{\it nw}-selective properties
introduced in \cite{BG}. In particular, we provide consistent uncountable
examples of such spaces and we define \textit{trivial} R-,H-, and M-{\it
nw}-selective spaces the ones with countable net weight having, additionally,
the cardinality and the weight strictly less then $cov({\cal M})$, $\frak b$,
and $\frak d$, respectively. Since we establish that spaces having
cardinalities more than $cov({\cal M})$, $\frak b$, and $\frak d$, fail to have
the R-,H-, and M-{\it nw}-selective properties, respectively, non-trivial
examples should eventually have weight greater than or equal to these small
cardinals. Using forcing methods, we construct consistent countable non-trivial
examples of R-{\it nw}-selective and H-{\it nw}-selective spaces and we
establish some limitations to constructions of non-trivial examples. Moreover,
we consistently prove the existence of two H-{\it nw}-selective spaces whose
product fails to be M-{\it nw}-selective. Finally, we study some relations
between {\it nw}-selective properties and a strong version of the HFD property.
