We first prove in this paper that a bounded weak $\kappa\overline{\theta }$-cover of any space has a $B(D,\omega_{0})$-refinement for any infinite cardinal number $\kappa$. The special case $\kappa=\aleph_{0}$ had already been proved by R.H.Price and J.C.Smith in \cite{PrSm}. Thus we obtain a characterization of $B(D,\omega_{0})$-refinability via bounded weak $\kappa\overline{\theta }$-cover refinements. We also prove that the set of all those points in any space having positive and finite order with respect to a given open family is covered by a $\sigma$-discrete closed refinement of that family. Thus a theorem of Bennett and Lutzer on subparacompactness is obtained as a corollary. We finally give a healthy proof of the fact that every weakly $\overline{\theta }$-refinable space is $B(D,\omega_{0}^{2})$-refinable.