Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes

Abstract

"The interplay between topological hyperconvex spaces and sigma-finite measures in such spaces gives rise to a set of analytical observations. This paper introduces the Noetherian class of kfinite k-hyperconvex topological subspaces (NHCs) admitting countable finite covers. A sigma-finite measure is constructed in a sigma-semiring in a NHC under a topological ordering of NHCs. The topological ordering relation maintains the irreflexive and anti-symmetric algebraic properties while retaining the homeomorphism of NHCs. The monotonic measure sequence in a NHC determines the convexity and compactness of topological subspaces. Interestingly, the topological ordering in NHCs in two isomorphic topological spaces induces the corresponding ordering of measures in sigma-semirings. Moreover, the uniform topological measure spaces of NHCs need not always preserve the pushforward measures, and a NHC semiring is functionally separable by a set of inner-measurable functions. ? 2022 by the author. Licensee MDPI, Basel, Switzerland.