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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.

03F35 Second- and higher-order arithmetic and fragments
03F60 Constructive and recursive analysis
26E40 Constructive real analysis
54E45 Compact (locally compact) metric spaces
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