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Compactness is one of the core notions of analysis: it connects localproperties to global ones and makes limits well-behaved. We study thecomputational properties of the compactness of Cantor space $2^{\mathbb{N}}$for uncountable covers. The most basic question is: how hard is it to compute afinite sub-cover from such a cover of $2^{\mathbb{N}}$? Another naturalquestion 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 fanfunctionals respectively compute such finite sub-covers and sequences. In thispaper, we establish the connection between these new fan functionals on onehand, and various well-known comprehension axioms on the other hand, includingarithmetical comprehension, transfinite recursion, and the Suslin functional.In the spirit of Reverse Mathematics, we also analyse the logical strength ofcompactness in Nonstandard Analysis. Perhaps surprisingly, the results in thelatter mirror (often perfectly) the computational properties of the special andweak fan functionals. In particular, we show that compactness (nonstandard orotherwise) 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).

The strength of compactness in Computability Theory and Nonstandard Analysis