Z-stability in Constructive Analysis

TL;DR

This paper introduces Z-stability within Bishop's constructive analysis, exploring its properties, limitations, and connections to constructive principles like the positivity principle and limited anti-Specker property.

Contribution

It defines Z-stability constructively, provides a recursive counterexample to classical uniformity arguments, and links it to key constructive principles.

Findings

Counterexample shows classical arguments fail constructively.

Positivity principle enables constructive uniformity results.

Connections established between anti-Specker property and Brouwer's fan theorem.

Abstract

We introduce Z-stability, a notion capturing the intuition that if a function f maps a metric space into a normed space and if the norm of f(x) is small, then x is close to a zero of f. Working in Bishop's constructive setting, we first study pointwise versions of Z-stability and the related notion of good behaviour for functions. We then present a recursive counterexample to the classical argument for passing from pointwise Z-stability to a uniform version on compact metric spaces. In order to effect this passage constructively, we bring into play the positivity principle, equivalent to Brouwer's fan theorem for detachable bars, and the limited anti-Specker property, an intuitionistic counterpart to sequential compactness. The final section deals with connections between the limited anti-Specker property, positivity properties, and (potentially) Brouwer's fan theorem for detachable bars.