Countable-density Hewitt–Marczewski–Pondiczery theorem

Arthur Freitas Ramos, David Barros Hulak, Ruy Jose Guerra Barretto de Queiroz

Palomar Registrypalomar/PALOMAR-2026-10-17-000019Oct 17, 2026math.GNscore 0

A product of at most continuum many separable topological spaces is separable. The proof constructs a countable family of finite binary tables that realizes every finite assignment of natural-number labels, then applies dense sequences in the factors. No separation, metrizability, compactness or countable-base assumption on the factors is needed. Empty factors and empty index sets are allowed. A companion theorem proves the sharp criterion for discrete Boolean cubes: Bool^I is separable exactly when the index cardinality is at most continuum. This is the classical countable-density case; no…

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