The relations among the notions of various kinds of stability and their applications

Abstract

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.