nLab Cauchy approximation

Contents

Contents

Definition

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

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

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

The set of all Cauchy approximations is defined as

C(S){xS +|δ +.η +.x(δ) δ+ηx(η)}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 xMx \circ M of a net xx and an modulus of convergence MM.

See also

References

Last revised on May 31, 2022 at 13:28:46. See the history of this page for a list of all contributions to it.