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.