The dimension of X n where X is a separable metric space

For a separable metric space X, we consider possibilities for the sequence S(X) = {dn : n ∈ N} where dn = dimX. In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is Xn such that S(Xn) = {n, n + 1, n + 2, . . .}, Yn, for n > 1, such that S(Yn) = {n, n+ 1, n+ 2, n+ 2, n+ 2, . . .}, and Z such that S(Z) = {4, 4, 6, 6, 7, 8, 9, . . .}. In Section 2, a subset X of R2 is shown to exist which satisfies 1 = dimX = dimX2 and dimX3 = 2. 0. Introduction and preliminaries. In this paper, we are concerned with problems related to the following question: Question. Suppose D = {dn : n ∈ N} is a sequence of positive integers. Under what conditions is there a separable metric space XD such that , for each n ∈ N, dimX D = dn? In case a sequence D has an XD, we say D is an allowable sequence and that XD realizes D. The sequence {kn : n ∈ N} is realized by X = I, but there are other allowable sequences. The well-known example of Erdős (see [E]) shows that the sequence {dn : n ∈ N} where each dn is 1 is allowable; Anderson and Keisler [AK] improved this, showing that each dn = k is allowable. In [Ku1], it is shown that, given m and k with k ≥ m, there is a sequence D where d1 = m and for all large enough n, dn = k. Obviously, if D is an allowable sequence, then D is nondecreasing, and for each n, dn+1 − dn ≤ d1, but not all sequences with these properties are allowable. For example, no sequence starting out as 1, 1, 2, 3 is allowable since if dimX2 = 1, then dimX4 = dim(X2)2 ≤ 2. We say a sequence D = {dn : n ∈ N} of positive integers is subadditive provided that, whenever s, t ∈ N, ds+t ≤ ds + dt. It is not hard to see that 1991 Mathematics Subject Classification: Primary 54F45, 54G20.