Open Research Index

Smirnov's local-to-global metrization theorem

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

Palomar Registrypalomar/PALOMAR-2026-10-15-000002Oct 15, 2026math.GNscore 0

A space is metrizable if and only if it is Hausdorff, paracompact and locally metrizable. A reusable construction gives a sigma locally finite basis from any point-finite open cover of metrizable patches in a paracompact space, without a separation hypothesis. Local neighborhoods, the cover index and each basis layer may have arbitrary cardinality. The proof assumes no second countability, separability, local compactness or nonemptiness. Classical mathematics; no novelty or priority claim.

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