Lebesgue numbers and Atsuji spaces in subsystems of second order   arithmetic
  We study properties of complete separable metric spaces within the framework of subsystems of second order arithmetic. In particular we consider Lebesgue and Atsuji spaces. The former are those such that every open covering U has a Lebesgue number, i.e. a positive number q such that for every point x of the space, there exists an element of U which contains the ball of center x and radius q; the latter are those such that every continuous function into another complete separable metric space is uniformly continuous.   The main results we obtain are the following: the statement "every compact space is Lebesgue" is equivalent to WKL_0; the statements "every perfect Lebesgue space is compact" and "every perfect Atsuji space is compact" are equivalent to ACA_0; the statement "every Lebesgue space is Atsuji" is provable in RCA_0; the statement "every Atsuji space is Lebesgue" is provable in ACA_0, but we do not know if it is equivalent to ACA_0. We also prove that the statement "the distance from a closed set is a continuous function" is equivalent to Pi^1_1-CA_0; the statements "there exists a complete separable metric space which is perfect and Heine-Borel compact (resp. Lebesgue, Atsuji)" are all equivalent to WKL_0. 
Mariagnese Giusto and Alberto Marcone