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A Solution of a Problem of I. P. Natanson Concerning the Decomposition of an Interval into Disjoint Perfect Sets
Formerly of the Information Sciences Branch, Naval Surface Warfare Center, White Oak, Silver Spring, MD, USA
- 1 Formerly of the Information Sciences Branch, Naval Surface Warfare Center, White Oak, Silver Spring, MD, USA
Advances in Pure Mathematics·Volume 04 (2014)·Pages 189–193·Published 5 May 2014·DOI10.4236/apm.2014.45024
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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.
KeywordsSpace-Filling CurvePerfect SetsInverse Image of a Perfect SetVertical Line Segments
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