  The paper is a contribution to intuitionistic reverse mathematics. We
introduce a formal system called Basic Intuitionistic Mathematics BIM, and then
search for statements that are, over BIM, equivalent to Brouwer's Fan Theorem
or to its positive denial, Kleene's Alternative to the Fan Theorem. The Fan
Theorem is true under the intended intuitionistic interpretation and Kleene's
Alternative is true in the model of BIM consisting of the Turing-computable
functions. The task of finding equivalents of Kleene's Alternative is,
intuitionistically, a nontrivial extension of finding equivalents of the Fan
Theorem, although there is a certain symmetry in the arguments that we shall
try to make transparent.
  We introduce closed-and-separable subsets of Baire space and of the set of
the real numbers. Such sets may be compact and also positively noncompact. The
Fan Theorem is the statement that Cantor space, or, equivalently, the unit
interval, is compact, and Kleene's Alternative is the statement that Cantor
space, or, equivalently, the unit interval is positively noncompact. The class
of the compact closed-and-separable sets and also the class of the
closed-and-separable sets that are positively noncompact are characterized in
many different ways and a host of equivalents of both the Fan Theorem and
Kleene's Alternative is found.
