Completeness of topological spaces: An induction-free review

TL;DR

This paper introduces an induction-free approach to defining completeness in topological spaces using graded bases and approach between nets, extending classical results to a broader class of spaces called lsb-spaces.

Contribution

It proposes a novel induction-free framework for completeness in base spaces, generalizing classical concepts to locally symmetric base spaces beyond uniform spaces.

Findings

Classical completeness results extend to lsb-spaces.

Characterization of compactness and Baire's theorem hold in this framework.

Existence of completions and completeness in product and function spaces are established.

Abstract

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 $X=(X,\tau)$, suppose we fix a base $\mathcal{B}$ of $\tau$ that is \emph{graded}, in the sense it is partitioned as $\mathcal{B}=\bigcup_{\varepsilon\in \mathcal{E}}\mathcal{B}_\varepsilon$ into open covers $\mathcal{B}_\varepsilon$ of $X$, making $X=(X,\tau,\mathcal{B})$ a \emph{(graded) base space}. If we now relax the notion of \emph{convergence of nets} to a notion of \emph{approach between nets} in $X$, then we obtain a more natural \emph{induction-free} notion of a \emph{cauchy net} in a base space, hence a corresponding \emph{induction-free} notion of \emph{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 \emph{locally symmetric base spaces} or \emph{$lsb$-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.