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

Edgar A. Cohen^{*}

Show more

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.

References

[1] W. Sierpinski, “Un Theoreme sur les Continus,” Tohoku Mathematical Journal, Vol. 13, 1918, pp. 300-305.

[2] G. F. Simmons, “Introduction to Topology and Modern Analysis,” McGraw Hill Book Company, Inc., New York, 1963, p. 92.

[3] I. P. Natanson, “Theory of Functions of a Real Variable,” Frederick Ungar Publishing Co., Inc., New York, 1961, pp. 49-50.

[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.