  We investigate the connections between computability theory and Nonstandard
Analysis. In particular, we investigate the two following topics and show that
they are intimately related.
  (T.1) A basic property of Cantor space $2^{\mathbb{N}}$ is Heine-Borel
compactness: For any open cover of $2^{\mathbb{N}}$, there is a finite
sub-cover. A natural question is: How hard is it to compute such a finite
sub-cover? We make this precise by analyzing the complexity of functionals that
given any $g:2^{\mathbb{N}}\rightarrow \mathbb{N}$, output a finite sequence
$\langle f_0 , \dots, f_n\rangle $ in $2^{\mathbb{N}}$ such that the
neighbourhoods defined from $\bar{f_i}g(f_i)$ for $i\leq n$ form a cover of
Cantor space.
  (T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham
Robinson's nonstandard compactness, i.e. that every binary sequence is
`infinitely close' to a standard binary sequence. We analyze the strength of
this nonstandard compactness property of Cantor space, compared to the other
axioms of Nonstandard Analysis and usual mathematics.
  The study of (T.1) gives rise to exotic objects in computability theory,
while (T.2) leads to surprising results in Reverse Mathematics. We stress that
(T.1) and (T.2) are highly intertwined and that our study of these topics is
`holistic' in nature: results in computability theory give rise to results in
Nonstandard Analysis and vice versa.
