It is proved first in this paper that all weakly 
$\kappa\overline{\theta}$-refinable spaces which 
were defined recently in [4] are irreducible for 
any infinite cardinal $\kappa$, i.e. any open 
cover of such spaces has a minimal open refinement. 
The special case $\kappa =\aleph _0$ has been proved 
before by J.C.Smith. A generalization of weakly 
$\kappa\overline{\theta}$-refinable and weakly 
$\overline{\delta\theta}$-refinable spaces is 
defined as weakly $\overline{\delta _{\kappa}\theta}$-
refinable and it is proved for $\kappa =\aleph _{\alpha}$ 
that any $\aleph _{\alpha +1}$-compact, weakly 
$\overline{\delta _{\kappa}\theta}$-refinable space 
has the Lindel\"of number $\le \aleph _{\alpha }$. 
Thus it is shown as a corollary that a regular, perfect, 
$\aleph _1$-compact $T_1$ space is hereditarily 
paracompact if it is weakly $\delta\theta$-refinable.