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
Cite this paper
Cohen Jr., E. (2014) A Solution of a Problem of I. P. Natanson Concerning the Decomposition of an Interval into Disjoint Perfect Sets. Advances in Pure Mathematics
, 189-193. doi: 10.4236/apm.2014.45024
 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
 Natanson, I.P. (1961) Theory of Functions of a Real Variable. Volume 1, Frederick Ungar Publishing Co., Inc., New York, 54.
 Rose, N.J. (2010) Hilbert-Type Space-Filling Curves. http://www4.ncsu.edu/~njrose/pdfFiles/HilbertCurve.pdf
 Hall, D.W. and Spencer II, G.L. (1955) Elementary Topology. John Wiley and Sons, Inc., New York, 72, 78.
 Bourbaki, N. (1971) General Topology. Springer-Verlag, 24.