In this paper we shall show: 
(1) Let $X$ be a zero-dimensional metric space and $Y$ be a $P$-space. 
If $Y$ has \textit{property} $B(D,\omega)$, 
then $X \times Y$ has \textit{property} $B(D,\omega)$.
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(2)  Let $X$ be a regular $\sigma$-space 
and $Y$ be a $P$-space. If $Y$ has \textit{property} $B(LF,\omega)$, 
then $X \times Y$ has \textit{property} $B(LF,\omega)$.
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(3) Let $X$ be a normal strong $\Sigma$-space and $Y$ be a $P$-space. 
If $Y$ has \textit{property} $B(LF,\omega)$, 
then $X \times Y$ has \textit{property} $B(LF,\omega)$.
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(4) Let $X$ be a strong $\Sigma$-space
 and $Y$ be a $P$-space. 
If $Y$ is weak $\overline{\theta}$-refinable 
(resp. weak $\overline{\delta\theta}$-refinable),
 then $X \times Y$ is weak $\overline{\theta}$-refinable.
 (resp. weak $\overline{\delta\theta}$-refinable).