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

Affiliation(s)

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

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

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.

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

Cohen Jr., E. (2014) A Solution of a Problem of I. P. Natanson Concerning the Decomposition of an Interval into Disjoint Perfect Sets.

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.

[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.