Contents

# Contents

## Definition

Let $\mathbb{Q}$ be the rational numbers and let

$\mathbb{Q}_{+} \coloneqq \{x \in \mathbb{Q} \vert 0 \lt x\}$

be the set of positive rational numbers. Let $(S, \sim)$ be a Booij premetric space. A Cauchy approximation is a function $x: \mathbb{Q}_{+} \to S$ such that for all positive rational numbers $\delta$ and $\eta$, $x(\delta) \sim_{\delta + \eta} x(\eta)$.

The set of all Cauchy approximations is defined as

$C(S) \coloneqq \{x \in S^{\mathbb{Q}_{+}} \vert \forall \delta \in \mathbb{Q}_{+}.\forall \eta \in \mathbb{Q}_{+}.x(\delta) \sim_{\delta + \eta} x(\eta)\}$

## Properties

Every Cauchy approximation is a Cauchy net indexed by the positive rational numbers $\mathbb{Q}_{+}$.

A Cauchy approximation is the composition $x \circ M$ of a net $x$ and an modulus of convergence $M$.