Lebesgue numbers and Atsuji spaces in subsystems of second-order arithmetic

Archive for Mathematical Logic 37 (5-6):343-362 (1998)
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Abstract

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to $\hbox{\sf WKL}_0$ ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to $\hbox{\sf ACA}_0$ ; the statement “every Lebesgue space is Atsuji” is provable in $\hbox{\sf RCA}_0$ ; the statement “every Atsuji space is Lebesgue” is provable in $\hbox{\sf ACA}_0$ . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to $\Pi^1_1\hbox{-\sf CA}_0$

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Citations of this work

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Located sets and reverse mathematics.Mariagnese Giusto & Stephen G. Simpson - 2000 - Journal of Symbolic Logic 65 (3):1451-1480.
Fundamental notions of analysis in subsystems of second-order arithmetic.Jeremy Avigad - 2006 - Annals of Pure and Applied Logic 139 (1):138-184.

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References found in this work

[Omnibus Review].Yiannis N. Moschovakis - 1968 - Journal of Symbolic Logic 33 (3):471-472.
Fixed point theory in weak second-order arithmetic.Naoki Shioji & Kazuyuki Tanaka - 1990 - Annals of Pure and Applied Logic 47 (2):167-188.

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