  A space $X$ has countable $(F)$-property if it has countable point network
satisfying the Collins-Roscoe structuring mechanism. Some sufficient conditions
for $C_p(X)$ having countable $(F)$-property are obtained. As a corollary, we
prove that if $X$ is Corson compact, $C_p(X)$ satisfies countable $(F)$. This
answers a question raised by Tkachuk. Also we get a class of function spaces
with hereditarily $D$-property. We also prove that the countable $(F)$-property
is preserved by taking $\Sigma_s$-product. This answers questions of Tkachuk
positively.
