All Title Author
Keywords Abstract


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, PP. 189-193

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

Full-Text   Cite this paper   Add to My Lib

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.

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.

Full-Text

comments powered by Disqus