We discuss some properties of topological groups with $D$-$sn$-network and $D$-$weak$-base, respectively, for a basic order $D$. We mainly give the following results: \begin{enumerate} ıem[(1)] Let $\kappa$ be an infinite cardinal and let $D$ and $E$ be directed sets. If $G$ is a topological group such that $G \leq_T D\times E$ and $t(G) = \lambda \leq \kappa$, then $G \leq_T E$. ıem[(2)] (CH) Let $G$ be a topological group, $H$ a normal closed subgroup of $G$ such that $H \leq_T \omega_1$ and $G/H \leq_T \omega^\omega$. Then $G \leq_T \omega^\omega$. ıem[(3)] Every Fréchet Hausdorff paratopological group $G$ having the property $(**)$ with a $D$-$sn$-network, for a basic order $D$, is first-countable. ıem[(4)] Let $D$ be a basic order. If a topological space $X$ has a $D$-$weak$-base, then it has countable cs$^*$-character.