%0 Journal Article
%T A Solution of a Problem of I. P. Natanson Concerning the Decomposition of an Interval into Disjoint Perfect Sets
%A Edgar A. Cohen Jr.
%J Advances in Pure Mathematics
%P 189-193
%@ 2160-0384
%D 2014
%I Scientific Research Publishing
%R 10.4236/apm.2014.45024
%X <p class=\"MsoNormal\" style=\"text-align:justify;\">
	<span lang=\"EN-US\"><span style=\"font-family:Verdana;\">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</span><i><span style=\"font-family:Verdana;\"> c</span></i><span style=\"font-family:Verdana;\"> 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.</span></span> 
</p>
%K Space-Filling Curve
%K Perfect Sets
%K Inverse Image of a Perfect Set
%K Vertical Line Segments
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=45778