Suppose that X=$\prod_{n<\omega}$X$_{n}$,if each space
$\prod_{i\leq n}$X$_{n}$ is $\delta\theta$-refinable
(i.e., submetalindelof), is X also $\delta\theta$-refinable?
K.Chiba asked in [1]. This paper first show that an inverse
limit theorem for $\delta\theta$-refinable spaces. Using this,
we obtain the result: Let X=$\prod_{\alpha\in\Lambda}$X$_{\alpha}$
be $|\Lambda|$- paracompact, X is $\delta\theta$-refinable
iff $\prod_{\alpha\in F}$X$_{\alpha}$ is $\delta\theta$-refinable
for each F$\in$[$\Lambda$]$^{<\omega}$. Then, the above problem
is answered positively. Next, we show that there are similar
results on hereditarily $\delta\theta$-refinable spaces.