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Compact Hausdorff countable-network metrization

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

Palomar Registrypalomar/PALOMAR-2026-10-17-000011Oct 17, 2026math.GNscore 0

For an arbitrary compact Hausdorff space, metrizability and second countability are each equivalent to the existence of a countable network of arbitrary subsets. A genuine Urysohn-function construction from pairs of network elements supplies continuous separation, and a compatible metric induces exactly the original topology. Supporting results give bounded-real-coordinate closed embeddings, preservation under continuous surjections, arbitrary-subspace heredity and compact-subspace metrization. Empty spaces and arbitrary universes are allowed. This is classical mathematics; no novelty or…

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