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

DOI: 10.4236/apm.2014.45024   PDF   HTML     4,445 Downloads   5,486 Views  

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.

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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, 4, 189-193. doi: 10.4236/apm.2014.45024.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] 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
[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
[4] Hall, D.W. and Spencer II, G.L. (1955) Elementary Topology. John Wiley and Sons, Inc., New York, 72, 78.
[5] Bourbaki, N. (1971) General Topology. Springer-Verlag, 24.

  
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