  We study closed choice principles for different spaces. Given information
about what does not constitute a solution, closed choice determines a solution.
We show that with closed choice one can characterize several models of
hypercomputation in a uniform framework using Weihrauch reducibility. The
classes of functions which are reducible to closed choice of the singleton
space, of the natural numbers, of Cantor space and of Baire space correspond to
the class of computable functions, of functions computable with finitely many
mind changes, of weakly computable functions and of effectively Borel
measurable functions, respectively. We also prove that all these classes
correspond to classes of non-deterministically computable functions with the
respective spaces as advice spaces. Moreover, we prove that closed choice on
Euclidean space can be considered as "locally compact choice" and it is
obtained as product of closed choice on the natural numbers and on Cantor
space. We also prove a Quotient Theorem for compact choice which shows that
single-valued functions can be "divided" by compact choice in a certain sense.
Another result is the Independent Choice Theorem, which provides a uniform
proof that many choice principles are closed under composition. Finally, we
also study the related class of low computable functions, which contains the
class of weakly computable functions as well as the class of functions
computable with finitely many mind changes. As one main result we prove a
uniform version of the Low Basis Theorem that states that closed choice on
Cantor space (and the Euclidean space) is low computable. We close with some
related observations on the Turing jump operation and its initial topology.
