Notre Dame Journal of Formal Logic

Classifying Dini's Theorem

Josef Berger and Peter Schuster

Abstract

Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the fan theorem, weak König's lemma, in the original classical setting of reverse mathematics started by Friedman and Simpson.

Article information

Source
Notre Dame J. Formal Logic, Volume 47, Number 2 (2006), 253-262.

Dates
First available in Project Euclid: 25 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1153858650

Digital Object Identifier
doi:10.1305/ndjfl/1153858650

Mathematical Reviews number (MathSciNet)
MR2240623

Zentralblatt MATH identifier
1156.03055

Subjects
Primary: 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30]
Secondary: 26E40: Constructive real analysis [See also 03F60] 54E45: Compact (locally compact) metric spaces

Keywords
compact metric spaces continuous functions uniform convergence reverse mathematics constructive mathematics

Citation

Berger, Josef; Schuster, Peter. Classifying Dini's Theorem. Notre Dame J. Formal Logic 47 (2006), no. 2, 253--262. doi:10.1305/ndjfl/1153858650. https://projecteuclid.org/euclid.ndjfl/1153858650


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