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

has been cited by the following article:

• TITLE: A Solution of a Problem of I. P. Natanson Concerning the Decomposition of an Interval into Disjoint Perfect Sets

  AUTHORS: Edgar A. Cohen Jr.

  KEYWORDS: Space-Filling Curve, Perfect Sets, Inverse Image of a Perfect Set, Vertical Line Segments

  JOURNAL NAME: Advances in Pure Mathematics, Vol.4 No.5, May 12, 2014

  ABSTRACT: 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.