Problem 909

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claims/: The 1 claim page of Problem 909, one per claimant's result; the problem's standing derives from them.

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Statement. Let $n\geq 2$. Is there a space $S$ of dimension $n$ such that $S^2$ also has dimension $n$?

Status. Proved, on the site's label, which credits Anderson and Keisler's 1967 construction for the general case; see the claim page.

Source. erdosproblems.com/909, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #909, https://www.erdosproblems.com/909.

References.

• [AnKe67] Anderson, R. D. and Keisler, J. E., An example in dimension theory. Proc. Amer. Math. Soc. 18 (1967), no. 4, 709--713, DOI 10.1090/S0002-9939-1967-0215288-0; the article is served free by the publisher.

Formalization. No statement file for the problem exists in formal-conjectures (none on main on 2026-10-07; the site's indicator reads "Formalised statement? No", and the community database records the problem proved and unformalized). A Lean formalization of Anderson and Keisler's result in Boris Alexeev's lean-proofs repository, naming R. D. Anderson and J. E. Keisler as its informal authors and Codex and GPT-5.6 Sol as its formal authors, is linked on the claim page; it was not built or audited here, and the standing rests on the refereed paper and the curator's credit.

Current assessment

Dated search scope (2026-10-07): the site's problem page, with its empty discussion thread and empty proof-claim tab, labels the problem proved and credits Anderson and Keisler for the general case; formal-conjectures has no statement file for the problem; the community database records the problem proved and unformalized, as of its last update, dated 31 August 2025; Boris Alexeev's lean-proofs repository holds the file src/latest/ErdosProblems/Erdos909.lean, added on 20 August 2026 and linked above at a fixed commit. Not searched: MathSciNet, zbMATH, Google Scholar.

Theorem 2 of [AnKe67] gives, for each $m\geq1$, a set $K\subset E^m$ with $\dim K=\dim K^s=\dim K^\omega=m-1$ for every positive integer $s$, where $\dim$ is the inductive dimension of Hurewicz and Wallman, $K^s$ the $s$-fold power and $K^\omega$ the countable power. Applied in $E^{n+1}$ it gives, for every $n\geq1$, a set of dimension $n$ whose finite and countable powers all have dimension $n$, so the answer is yes for every $n\geq2$; this is the claim page. The paper 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, which is why the construction is needed. The set is built by transfinite induction over the nondegenerate continua of $E^m$; the proofs (Lemmas 1--4 and Theorems 1 and 2) are not reconstructed in this corpus.

Known Results

The statements of Theorems 1 and 2 and the lemmas of [AnKe67] are recorded on the source card.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• anderson_1967_example_dimension_theory
• anderson_1967_example_dimension_theory / theorem_1
• anderson_1967_example_dimension_theory / theorem_2