  First, we prove that a random metric space can be isometrically embedded into
a complete random normed module, as an application of which, it is easy to see
that the notion of $d$-$\sigma$-stability introduced for a nonempty subset of a
random metric space can be regarded as a special case of the notion of
$\sigma$-stability introduced for a nonempty subset of a random normed module,
as another application we give the final version of the characterization for a
$d$-$\sigma$-stable random metric space to be stably compact. Second, we prove
that an $L^{\infty}$-module is an $L^{p}$-normed $L^{\infty}$-module iff it is
generated by a complete random normed module, from which it is easily seen that
the gluing property of an $L^{p}$-normed $L^{\infty}$-module can be derived
from the $\sigma$-stability of the generating random normed module, as
applications the known and new basic facts of module duals for $L^{p}$-normed
$L^{\infty}$-modules can be obtained, in a simple and direct way, from the
theory of random conjugate spaces of random normed modules. Third, we prove
that a random normed space is order complete iff it is complete with respect to
the $(\varepsilon,\lambda)$-topology, as an application it is proved that the
$d$-decomposability of an order complete random normed space is exactly its
$d$-$\sigma$-stability. Finally, we prove that an equivalence relation on the
product space $X\times B$ of a nonempty set $X$ and a complete Boolean algebra
$B$ is regular iff it can be induced by a $B$-valued Boolean metric $d$ on $X$,
as an application it is proved that a nonempty subset of a Boolean set $(X,d)$
is universally complete iff it is a $B$-stable set defined by a regular
equivalence relation.
