  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.
