A dimensional property of Cartesian product

We show that the Cartesian product of three hereditarily infinite dimensional compact metric spaces is never hereditarily infinite dimensional. It is quite surprising that the proof of this fact (and this is the only proof known to the author) essentially relies on algebraic topology.

💡 Research Summary

The paper investigates a subtle question in dimension theory: whether the Cartesian product of several hereditarily infinite‑dimensional compact metric spaces retains the property of being hereditarily infinite‑dimensional. A space X is called hereditarily infinite‑dimensional if every non‑empty closed subset of X has infinite covering dimension (dim X = ∞). While individual examples of such spaces are known and have been studied extensively through cohomological dimension, extension theory, and related tools, the behavior of their products has remained largely mysterious. The author proves a striking negative result: the product of three hereditarily infinite‑dimensional compact metric spaces can never be hereditarily infinite‑dimensional. In other words, once we take a triple product, the resulting space inevitably contains a non‑empty closed subset of finite covering dimension, thereby violating the hereditary infinite‑dimensional condition.

The proof is notable for its reliance on algebraic topology, specifically on cohomological dimension theory, Bockstein spectral sequences, and Steenrod operations. The argument proceeds in several stages. First, the author recalls the necessary background: the definition of covering dimension, its cohomological counterpart dim_G X (the smallest n such that H^{n+1}(X; G)=0 for a given coefficient group G), and the relationship between hereditary infinite‑dimensionality and infinite cohomological dimension for all non‑trivial coefficient groups. The paper also reviews Dranishnikov’s extension dimension theory, which connects the ability to extend maps into a CW‑complex K with the inequality dim_K X ≤ n.

Next, the author selects three arbitrary hereditarily infinite‑dimensional compact metric spaces X, Y, Z and examines their product W = X × Y × Z. By choosing a prime p and working with coefficients in ℤ/pℤ, the author shows that each factor has non‑trivial cohomology in arbitrarily high degrees. Using the Künneth formula for Čech cohomology, the product W inherits non‑trivial cohomology classes in a range of degrees. Crucially, the author applies Steenrod squares Sq^i to these classes, which raise the degree while preserving non‑triviality under the conditions guaranteed by the hereditary infinite‑dimensionality of the factors. This produces a non‑zero cohomology class in a specific degree n that survives under the Bockstein homomorphism, yielding a non‑trivial element in H^n(W; ℤ/pℤ).

Having identified such a class, the author invokes the extension dimension machinery: the existence of a non‑zero class in H^n(W; ℤ/pℤ) forces the extension dimension of W to be bounded above by the n‑dimensional Moore space M(ℤ/pℤ, n). Consequently, any closed subset of W that carries this class must have covering dimension at most n. This directly contradicts the definition of hereditary infinite‑dimensionality, which would require every closed subset to have infinite dimension. Hence the initial assumption—that the triple product remains hereditarily infinite‑dimensional—must be false.

The paper concludes with several remarks. First, the result demonstrates that hereditary infinite‑dimensionality is not preserved under triple products, a phenomenon that was unexpected given the stability of many other dimension‑related properties under products. Second, the proof showcases how tools from algebraic topology—particularly cohomology operations—can resolve problems that appear purely metric‑topological. Finally, the author points out open problems: the status of double products remains unresolved (it is unknown whether X × Y can be hereditarily infinite‑dimensional), and the possibility of extending the method to higher products or to non‑compact settings is suggested as a direction for future research. Overall, the paper provides a deep and technically sophisticated answer to a natural question in infinite‑dimensional topology, enriching our understanding of how algebraic invariants control the geometry of product spaces.