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 k-finite 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.

Homotopy of Linearly Ordered Split–Join Chains in Covering Spaces of Foliated n-Manifold Charts

“…However, this is a stronger condition for retaining ordering relations and it can be relaxed. Earlier, it was shown that two topological hyperconvex spaces can be ordered under a special topological-ordering relation affecting the measurability in such spaces [9]. A topological space X is Bolzano-Weierstrass space (B w − space) if every infinite subset of X has an accumulation point.…”