In a
previous paper published in this journal, it was demonstrated that any bounded,
closed interval of the real line can, except for a set of Lebesgue measure 0,
be expressed as a union of c pairwise
disjoint perfect sets, where c is the cardinality of the continuum. It turns
out that the methodology presented there cannot be used to show that such an
interval is actually decomposable into c nonoverlapping perfect sets without
the exception of a set of Lebesgue measure 0. We shall show, utilizing a
Hilbert-type space-filling curve, that such a decomposition is possible.
Furthermore, we prove that, in fact, any interval, bounded or not, can be so
expressed.

Cohen Jr., E.A. (2013) On the Decomposition of a Bounded Close Interval of the Real Line into Closed Sets. Advances in Pure Mathematics, 3, 405-408. http://dx.doi.org/10.4236/apm.2013.34058