2026/03/04 by Earnest Akofor · 1 voice
Mathematics · #math.GN
Completeness for a (topological) space is often based on the existence of special structures (such as metrics, uniformities, proximities, convergences, etc) that explicitly induce the topology, making the completeness induction-dependent. However, in any given space X=(X,τ), suppose we fix a base B of τ that is graded, in the sense it is partitioned as B=\bigcupε∈ EBε into open covers Bε of X, making X=(X,τ,B) a (graded) base space. If we now relax the notion of convergence of nets to a notion of approach between nets in X, then we obtain a more natural induction-free notion of a cauchy net in a base space, hence a corresponding induction-free notion of completeness for base spaces. We find that many classical concepts and results on completeness for uniform spaces carry over to completeness for a certain class of base spaces (named locally symmetric base spaces or lsb-spaces) that properly contains uniform spaces. The said classical results include characterization of compactness, Baire's theorem, existence of a completion, and completeness results for product and function lsb-spaces.