Beta Spaces: New Generalizations of Typically-Metric Properties

TL;DR

This paper introduces $eta$-spaces, a new class of topological spaces that generalize metric properties and uniform spaces, enabling broader definitions of completeness and boundedness, and extending key theorems like the Contraction Mapping Theorem.

Contribution

The paper constructs $eta$-spaces, providing a new framework that generalizes uniform spaces and allows for the extension of metric theorems to more general topological spaces.

Findings

$eta$-spaces are strictly more general than uniform spaces

Generalized definitions of completeness and boundedness in $eta$-spaces

Extended versions of metric theorems, including the Contraction Mapping Theorem

Abstract

It is well-known that point-set topology (without additional structure) lacks the capacity to generalize the analytic concepts of completeness, boundedness, and other typically-metric properties. The ability of metric spaces to capture this information is tied to the fact that the topology is generated by open balls whose radii can be compared. In this paper, we construct spaces that generalize this property, called $\beta$-spaces, and show that they provide a framework for natural definitions of the above concepts. We show that $\beta$-spaces are strictly more general than uniform spaces, a common generalization of metric spaces. We then conclude by proving generalizations of several typically-metric theorems, culminating in a broader statement of the Contraction Mapping Theorem.