We prove that a space with a $\sigma$-weakly hereditarily closure-preserving $sn$-network is $sn$-first countable. As an application of this result, we prove that a Lindelöf space with a $\sigma$-weakly hereditarily closurepreserving $sn$-network is $sn$-second countable. The above results answer some questions posed by L. Yan.