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

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 -CA0.