On the Decomposition of a Bounded Closed Interval of the Real Line into Closed Sets

Edgar A. Cohen

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

DOI: 10.4236/apm.2013.34058                

Abstract

It has been shown by Sierpinski that a compact, Hausdorff, connected topological space (otherwise known as a continuum) cannot be decomposed into either a finite number of two or more disjoint, nonempty, closed sets or a countably infinite family of such sets. In particular, for a closed interval of the real line endowed with the usual topology, we see that we cannot partition it into a countably infinite number of disjoint, nonempty closed sets. On the positive side, however, one can certainly express such an interval as a union of c disjoint closed sets, where c is the cardinality of the real line. For example, a closed interval is surely the union of its points, each set consisting of a single point being closed. Surprisingly enough, except for a set of Lebesgue measure 0, these closed sets can be chosen to be perfect sets, i.e., closed sets every point of which is an accumulation point. They even turn out to be nowhere dense (containing no intervals). Such nowhere dense, perfect sets are sometimes called Cantor sets.

Keywords

Cantor Sets; Cardinality; Disjoint Closed Sets; Dyadic Representation; Interlacing Dyadic Expansions; Generators of Sets; Nowhere Dense

Conflicts of Interest

The authors declare no conflicts of interest.

References

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[4]	D. W. Hall and G. L. Spencer II, “Elementary Topology,” John Wiley and Sons, Inc., New York and Chapman & Hall, Limited, London, 1955, p. 44.
[5]	M. E. Munroe, “Introduction to Measure and Integration,” Addison-Wesley Publishing Co., Inc., Reading, 1953, p. 70.