  Compactness is one of the core notions of analysis: it connects local
properties to global ones and makes limits well-behaved. We study the
computational properties of the compactness of Cantor space $2^{\mathbb{N}}$
for uncountable covers. The most basic question is: how hard is it to compute a
finite sub-cover from such a cover of $2^{\mathbb{N}}$? Another natural
question is: how hard is it to compute a sequence that covers $2^{\mathbb{N}}$
minus a measure zero set from such a cover? The special and weak fan
functionals respectively compute such finite sub-covers and sequences. In this
paper, we establish the connection between these new fan functionals on one
hand, and various well-known comprehension axioms on the other hand, including
arithmetical comprehension, transfinite recursion, and the Suslin functional.
In the spirit of Reverse Mathematics, we also analyse the logical strength of
compactness in Nonstandard Analysis. Perhaps surprisingly, the results in the
latter mirror (often perfectly) the computational properties of the special and
weak fan functionals. In particular, we show that compactness (nonstandard or
otherwise) readily brings us to the outer edges of Reverse Mathematics (namely
$\Pi_2^1$-CA$_0$), and even into Schweber's higher-order framework (namely
$\Sigma_{1}^{2}$-separation).
