Doesn't the standard proof of "Sierpinski's Theorem" work in ZF?
 
Where is choice used in the proofs below? Could it be easily eliminated if the sets $X_i$ were assumed to be singletons?

"Continuum" below means compact connected Hausdorff.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/ylcLe.png