Lebesgue numbers and Atsuji spaces in subsystems of secondorder arithmetic (1998)
| Venue: | Arch. Math. Logic |
| Citations: | 3 - 1 self |
BibTeX
@ARTICLE{Giusto98lebesguenumbers,
author = {Mariagnese Giusto and Alberto Marcone},
title = {Lebesgue numbers and Atsuji spaces in subsystems of secondorder arithmetic},
journal = {Arch. Math. Logic},
year = {1998}
}
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Abstract
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 WKL0; the statements “every perfect Lebesgue space is compact ” and “every perfect Atsuji space is compact ” are equivalent to ACA0; the statement “every Lebesgue space is Atsuji ” is provable in RCA0; the statement “every Atsuji space is Lebesgue ” is provable in ACA0. We also prove that the statement “the distance from a closed set is a continuous function ” is equivalent to Π 1 1-CA0.
