\(C(\infty,\infty)\):  The Axiom of Choice: Every  set  of  non-empty sets has a choice function.

There is a partition of the real line into \(\aleph_1\) Borel sets \(\{B_\alpha: \alpha<\aleph_1\}\) such that for some \(\beta <\aleph_1\), \(\forall\alpha <\aleph_1\), \(B_{\alpha}\in G_{\beta}\). (\(G_\beta\) for \(\beta < \aleph_1\) is defined by induction, \(G_0=\{A: A\) is an open subset of \({\Bbb R}\}\) and for \(\beta > 0\),

• \(G_\beta =\left\{\bigcup^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists\xi <\beta)(A_i\in G_\xi)\,\right\}\) if \(\beta\) is even and
• \(G_\beta = \left\{\bigcap^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists \xi < \beta)(A_{i}\in G_\xi)\,\right\}\) if \(\beta\) is odd.)