General and Set-Theoretic Topology
‟A quantitative measure of non-Isbell convexity in T_0-quasi-metric spaces” by Collins Amburo Agyingi <agyingic@gmail.com>, University of South Africa
‟A quantitative measure of non-Isbell convexity in T_0-quasi-metric spaces” by Collins Amburo Agyingi <agyingic@gmail.com>, University of South Africa
Abstract:
Isbell-convex (q-hyperconvex) T0-quasi-metric spaces are known to be bicomplete, although the converse does not hold in general. This talk introduces a quantitative invariant that measures the extent to which a bicomplete T0-quasi-metric space fails to be Isbell-convex. The invariant is defined as a two-component parameter
hq(X)=((hq)1(X),(hq)2(X)) capturing the inherently asymmetric forward and backward deviations from convexity.
Using the framework of minimal function pairs arising in the Isbell hull, we show that hq(X)=(0,0) characterizes Isbell-convexity. We further establish key properties of this invariant, including non-negativity, invariance under isometries, and monotonicity with respect to subspaces and nonexpansive maps, as well as boundedness in the finite case.
Concrete examples demonstrate how bicomplete spaces may fail to be Isbell-convex and illustrate how the invariant quantifies this deviation. This provides a new quantitative tool for studying the interplay between convexity and completeness in asymmetric topology.