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.