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.