Douglas Bridges ; James Dent ; Maarten McKubre-Jordens - Z-stability in Constructive Analysis

lmcs:2031 - Logical Methods in Computer Science, April 27, 2017, Volume 12, Issue 3 -
Z-stability in Constructive AnalysisArticle

Authors: Douglas Bridges ; James Dent ; Maarten McKubre-Jordens

    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.

    Volume: Volume 12, Issue 3
    Published on: April 27, 2017
    Accepted on: September 14, 2016
    Submitted on: February 1, 2016
    Keywords: Mathematics - Logic,03F60

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