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Claim. For every integer $m\geq1$ there is a set $K\subset E^m$, Euclidean $m$-space, such that $K$, every finite power $K^s$ and the countable power $K^\omega$ all have topological dimension $m-1$; here $\dim$ is the inductive dimension of Hurewicz and Wallman, and the sets are separable metric spaces as subspaces of $E^m$. This is Theorem 2 of Anderson and Keisler's paper. Taking $m=n+1$ gives, for every $n\geq1$ and so for every $n\geq2$, a space $S=K$ of dimension $n$ with $\dim S^2=n$, which answers the question of Problem 909 in the affirmative. The paper also notes the easy cases, stated here with $n$ the dimension: in dimension $0$, a Cantor set or the rationals; in dimension $1$, the rational points of Hilbert space, which the paper admits by relaxing its requirement $K\subset E^n$ to $K\subset E^{n+1}$; in dimension at least $2$, the standard examples contain cells, whose finite products increase in dimension. The paper's digest, with the statements of Theorems 1 and 2 and the lemmas, is the [[../library/analysis/anderson_1967_example_dimension_theory/_index|source card]]; its proofs are not reconstructed in this corpus.

Acceptance. The result is refereed: R. D. Anderson and J. E. Keisler, An example in dimension theory, Proc. Amer. Math. Soc. 18 (1967), no. 4, 709–713, received by the editors on December 16, 1966. It is reviewed in the sense of a documented independent acceptance: Thomas Bloom, the curator of erdosproblems.com, marks Problem 909 proved and credits Anderson and Keisler for the general case.

Formalization. The file src/latest/ErdosProblems/Erdos909.lean of Boris Alexeev's lean-proofs repository, linked above at a fixed commit and added on 20 August 2026, declares itself a formalization of a solution to the problem, naming R. D. Anderson and J. E. Keisler as its informal authors, Codex and GPT-5.6 Sol as its formal authors, and the 1967 paper as its primary reference. Its theorem erdos_909 states that for every natural number $n\ge2$ there is a topological space $S$ whose small inductive dimension is $n$ and whose square $S\times S$ also has small inductive dimension $n$; the file and the modules it imports contain no sorry. This corpus has not built or audited it, so no formalized evidence is listed; the standing rests on the refereed paper and the curator's credit. No statement file for the problem exists in formal-conjectures (none on main on 2026-10-07).

The page is dated by the paper's presentation to the American Mathematical Society on January 24, 1967, the earliest public date the paper prints; the issue appeared in August 1967.