TITLE:
A Solution of a Problem of I. P. Natanson Concerning the Decomposition of an Interval into Disjoint Perfect Sets
AUTHORS:
Edgar A. Cohen Jr.
KEYWORDS:
Space-Filling Curve, Perfect Sets, Inverse Image of a Perfect Set, Vertical Line Segments
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.4 No.5,
May
12,
2014
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.