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

Edgar A Cohen Jr

doi:10.4236/apm.2014.45024

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.

Highlights

In a paper published previously in this journal, it was shown that any bounded, closed interval of the real line can, except for a set of Lebesgue measure 0, be decomposed into c pairwise disjoint perfect sets, where c is the power of the continuum
Any bounded closed interval [a,b] of the real line is expressible as a union of c pairwise disjoint perfect sets
It has been shown that any subinterval of the real line can, first of all, be expressed as a union of c pairwise disjoint perfect sets in the subinterval relative topology

Summary

Introduction

In a paper published previously in this journal (see [1]), it was shown that any bounded, closed interval of the real line can, except for a set of Lebesgue measure 0, be decomposed into c pairwise disjoint perfect sets, where c is the power of the continuum. Any bounded closed interval [a,b] of the real line is expressible as a union of c pairwise disjoint perfect sets. One can map [0,1] continuously onto the closed square [0,1] × [0,1] of the xy-plane by invoking a Hilbert-type space-filling curve (see [3]).

Results

Conclusion