Anderson 1967 example dimension theory

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theorem_1: Anderson and Keisler's single-exponent construction: for positive integers n and s there is a set K in Euclidean n-space such that K and its s-fold power both have inductive topological dimension n minus one.

theorem_2: Anderson and Keisler's main theorem: there is a set K in Euclidean n-space such that K, every finite power of K and the countable power of K all have inductive topological dimension n minus one.

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R. D. Anderson and J. E. Keisler, An example in dimension theory, Proc. Amer. Math. Soc. 18 (1967), no. 4, 709--713; DOI 10.1090/S0002-9939-1967-0215288-0 (volume, issue and DOI from the Crossref record; the page heads show only "1967" and "August"). Presented to the Society January 24, 1967; received by the editors December 16, 1966.

The copy read for this card is a publisher scan of the five printed pages 709--713 with a machine text layer (physical PDF p. $n$ is printed p. $708+n$); the last page also carries the start of the following article in the issue. The text layer garbles most formulas, so the statements below were checked on the page images. Provenance: downloaded in the repository's survey of September 2026; the download URL was not recorded; 498,284 bytes. No notice is printed in the scan's text layer on pp. 709--710 or 712--713; the publisher's article page shows "(c) Copyright 1967 American Mathematical Society" (https://pubs.ams.org/journals/proc/1967-018-04/S0002-9939-1967-0215288-0, read 2026-10-02), every other right reserved.

Read status: claims checked. Theorems 1 and 2 were read clause by clause on the page images of pp. 709 and 712; the lemmas of section II and the proofs of section III were not read beyond their statements.

Contents

Throughout, $\dim$ is the (inductive) topological dimension of Hurewicz and Wallman, $E^n$ is Euclidean $n$-space, $K^s$ is the product of $s$ copies of $K$ and $K^\omega$ the product of countably many copies.

• Theorem 2 (stated p. 709 and p. 712; proof pp. 712--713): for the given $n$ (the statement leaves $n$ unquantified; the construction is for "arbitrary $n$", p. 709) one set $K\subset E^n$ serves all powers at once: $K$, each finite power $K^s$ ($s\ge1$) and the countable power $K^\omega$ have dimension $n-1$. Page 709 sets this against a result it calls known: when $A$ and $B$ are nonvoid separable metric spaces, $A$ compact and $\dim B>0$, then $\dim(A\times B)\ge\dim A$, with equality only if $\dim A=\infty$. The page also lists the easy cases: for $n=1$, a Cantor set or the rationals of the line; for $n=2$, if $K$ need not lie in $E^n$ (or need only lie in $E^{n+1}$), "the rationals in Hilbert space"; for $n>2$ the usual $n$-dimensional examples (Hurewicz and Wallman, pp. 29 and 64) contain cells, so their finite powers gain dimension.
• Theorem 1 (p. 712; proof p. 712): given positive integers $n$ and $s$ (the paper's $\omega$ is the set of positive integers, p. 709), some $K\subset E^n$ has $K$ and $K^s$ both of dimension $n-1$. The set is built by transfinite induction over the nondegenerate continua of $E^n$, well-ordered so that each has fewer than $\mathfrak{c}$ predecessors: each continuum the set does not yet meet receives one of its points, chosen so that the $s$-letter words over $K$ avoid a countable family of spheres in $E^{ns}$; Theorem 2 follows by a slight variant of this procedure that handles all $s$ at once.
• Lemmas 1--4 (pp. 710--711): Lemma 1, a set $K\subset E^n$ meeting every nondegenerate continuum has $\dim K\ge n-1$; Lemma 2, hyperplanes can be tilted into general position with respect to countably many families while still separating spheres (a weakened form is used "without explicit proof here", p. 710); Lemma 3, a set $T\subset E^{ns}$ missing the chosen spheres has $\dim T\le n-1$; Lemma 4, for $K\subset E^n$, if $\dim K^s<t$ for every $s$ then $\dim K^\omega<t$.

Compiled scope

Only the statements of Theorems 1 and 2 and the summary of the lemmas above were read. The proofs were not checked, and nothing here is independently reviewed.

Bears on. #909, which asks, for $n\ge2$, for a space of dimension $n$ whose square has dimension $n$: Theorem 2 applied in $E^{n+1}$ gives a set $K\subset E^{n+1}$ with $\dim K=\dim K^2=n$ for every $n\ge1$, and so such a space for every $n\ge2$; Theorem 1 with $s=2$ gives the same for each $n$ separately. The paper does not mention the problem.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.