The relations among the notions of various kinds of stability and their applications. (English) Zbl 07871743
Summary: First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of \(d\)-\(\sigma\)-stability in a random metric space can be regarded as a special case of the notion of \(\sigma\)-stability in 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 \(L^p\)-normed \(L^{\infty}\)-module is exactly generated by a complete random normed module so 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 direct relation between module duals and random conjugate spaces are given. Third, we prove that a random normed space is order complete iff it is \((\varepsilon,\lambda)\)-complete, 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 of a nonempty set \(X\) and a complete Boolean algebra \(B\) is regular iff it can be induced by a \(B\)-valued Boolean metric 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.
MSC:
| 46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |
| 18F15 | Abstract manifolds and fiber bundles (category-theoretic aspects) |
| 46A16 | Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
Keywords:
\(\sigma\)-stability; \(d\)-\(\sigma\)-stability; gluing property; \(d\)-decomposability; universal completeness; \(B\)-stabilityReferences:
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