The dimension of $X^n$ where $X$ is a separable metric space
Volume 150 / 1996
Fundamenta Mathematicae 150 (1996), 43-54
DOI: 10.4064/fm-150-1-43-54
Abstract
For a separable metric space X, we consider possibilities for the sequence $S(X) = {d_n: n ∈ ℕ}$ where $d_n = dim X^n$. 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 $X_n$ such that $S(X_n) = {n, n+1, n+2,...}$, $Y_n$, for n >1, such that $S(Y_n) = {n, n+1, n+2, n+2, n+2,...}$, and Z such that S(Z) = {4, 4, 6, 6, 7, 8, 9,...}. In Section 2, a subset X of $ℝ^2$ is shown to exist which satisfies $1 = dim X = dim X^2$ and $dim X^3 = 2$.
