In this paper we shall show: (1) Let $X$ be a zero-dimensional metric space 
and $Y$ be a $\delta$$\theta$-refinable $P$-space.
Then $X \times Y$ is $\delta$$\theta$-refinable.
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(2) Let $X$ be an almost expandable strong $\Sigma$-space and $Y$ 
be a strong $\delta$$\theta$-refinable $P$-space. 
Then $X \times Y$ is $\delta$$\theta$-refinable.
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(3) Let $X$ be a metrizable space and $Y$ be a w-$\delta$$\theta$-refinable $P$-space. 
Then $X \times Y$ is w-$\delta$$\theta$-refinable.
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Similar results of (3) for analogous properties also hold.