×

zbMATH — the first resource for mathematics

Classifying Dini’s theorem. (English) Zbl 1156.03055
Summary: 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.

MSC:
03F35 Second- and higher-order arithmetic and fragments
03F60 Constructive and recursive analysis
26E40 Constructive real analysis
54E45 Compact (locally compact) metric spaces
PDF BibTeX XML Cite
Full Text: DOI