Decompositions and dimension of closed sets in 𝑅ⁿ
Arthur N. Milgram · Transactions of the American Mathematical Society · 1938
Introduction.It is our purpose to give a characterization of the dimension of closed sets immersed in R* (euclidean space of » dimensions).This is done in terms of certain properties of the decompositions of these sets into a countable infinity of closed sets.The results are well known for finite decompositions of compact sets, but have never been shown to be so intrinsic a property of dimension as to remain valid under countable decompositions.This may be due to the fact that the proof seems to require much of the technique and many of the results of Alexandroff,f which are of very recent development.We may state our principal result as follows: A closed subset F of Rn is of dimension r if and only if there exists an « >0, such that F may be decomposed into the sum of a countable infinity of closed sets Fi,F2, ■ ■ ■ , F" ■ ■ ■ of diameter less than t,for which dim Fi-Ff&r-l, i^j, but for any such decomposition there exists a pair of integers m and » such that dim FmFn = r -l.This result follows quite readily when we have proved the following : If F is a closed subset of Rn, p a point of F, Fh F2, ■ ■ ■ , Fs, ■ ■ ■ a decomposition of F into closed sets, zn_r-1 a cycle in Sip, e) -F, which does not bound in Sip, «) -P but does bound in 5(^, e) -P.-, i = 1,2, • -• , then there exists a pair of integers m and « such that dim Fm-Fn-S(p, e)^r -l.From this we obtain an interesting result which may be considered a generalization of a theorem due to Miss Mullikin.tWe show that the sum of a countable number of closed sets, no one of which separates Rn, and the dimension of whose intersections taken pairwise does not exceed « -3, cannot separate R".The author takes this opportunity to express his gratitude to Professor J. R. Kline, whose suggestions and unfailing encouragement made this paper possible.2. Notation.The notation and definitions used in the sequel are widely employed.For example, 8(M) refers to the diameter of a point set M, p(Mi, Ms) to the distance between the sets Mi and M2, S(M, e) to the set of points x such that p(M, x)