Take the 2-minute tour ×  An anonymous question from the
20-questions seminar:  Can you explicitly write R^2 as a
disjoint union of two totally path disconnected sets?
share|improve this question I'd be happy with a non-explicit way
to make such a partition. –  Anton Geraschenko Oct 6 '09 at
23:57 Here's a nonexplicit construction. Enumerate (well-order)
all possible paths with order-type the continuum. By induction,
put one point from each path in each of the sets of the
partition. This is possible since each path contains continuum
many points and at any stage of the induction, you've only
chosen where less than continuum points go. –  Eric Wofsey Oct 9
'09 at 2:32 @Eric: that's awesome! What else could you possibly
do? –  Anton Geraschenko Oct 9 '09 at 20:17 What does "well-
order with order-type the continuum"? Maybe "biject the set of
paths with the first uncountable ordinal" so that all initial
segments of the ordinal are numerable? –  Mariano Suárez-Alvarez
Nov 18 '09 at 6:52 Mariano: No, you are assum