Recently someone asked on sci.math.research
whether R^3 could be foliated by straight lines,
with exactly one line in each direction. Someone
else soon provided a proof that this is impossible
(using homotopy theory).
If we drop the continuity requirement for a
foliation, the problem becomes this one:
QUESTION: Is R^3 the disjoint union of
[bi-infinite] straight lines L_d such that each
direction d occurs exactly once?
(I.e., each line through the origin of R^3 is
parallel or equal to exactly one of the L_d's.)
My guess is yes, but it's just a guess.
--Dan