In this paper, the followings are proved that: 
(1) Let X be the inverse limit of an inverse system 
$\{$X$_{\alpha},{\pi}^{\alpha}_{\beta},\Lambda\}$ and
let the projection ${\pi}_{\alpha}$ be an open 
and onto map for each ${\alpha}\in\Lambda$, 
if X is $|\Lambda|$-paracompact (resp. hereditarily $|\Lambda|$-paracompact) 
and each X$_{\alpha}$ has property ${\cal P}$ (resp. hereditarily property ${\cal P}$), 
then X has also property ${\cal P}$ (resp. hereditarily property ${\cal P}$). 
(2) Let X=$\prod_{\sigma\in\Sigma}$X$_{\sigma}$ be $|\Sigma|$-paracompact 
(resp. hereditarily $|\Sigma|$-paracompact), then X has property ${\cal P}$ 
(resp. hereditarily property ${\cal P}$) iff $\prod_{\sigma\in F}$X$_{\sigma}$ 
has property ${\cal P}$(resp. hereditarily property ${\cal P}$) 
for each F$\in$[$\Sigma$]$^{<\omega}$, where ${\cal P}$ denotes 
one of the following four properties:  expanability,   
discrete expandability, $\sigma$-expandability, discrete $\sigma$-expandability.