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Sierpiński's characterization of the rationals

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

Palomar Registrypalomar/PALOMAR-2026-10-23-000003Oct 23, 2026math.GNscore 0

Every nonempty countable metrizable topological space without isolated points is homeomorphic to the rational numbers with their usual topology. Conversely, these four properties characterize spaces homeomorphic to the rationals. A clopen basis is derived from the hypotheses. Scheduled binary clopen splits give each point an address with both bits arbitrarily far out; the induced lexicographic order is proved dense, without endpoints, and to generate exactly the original topology. The countable dense-order isomorphism theorem then gives the homeomorphism. This formalizes classical…

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