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

Edgar A. Cohen Jr.^*

Formerly of the Information Sciences Branch, Naval Surface Warfare Center, White Oak, Silver Spring, MD, USA.

DOI: 10.4236/apm.2014.45024                

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.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

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[2]	Natanson, I.P. (1961) Theory of Functions of a Real Variable. Volume 1, Frederick Ungar Publishing Co., Inc., New York, 54.
[3]	Rose, N.J. (2010) Hilbert-Type Space-Filling Curves. http://www4.ncsu.edu/~njrose/pdfFiles/HilbertCurve.pdf
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