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        "description": "The exact 1925 target is the normal, second-countable Hausdorff-space criterion in printed 310 Hauptsatz and its countable-separator metric construction, not an unqualified attribution of the later familiar regular-second-countable theorem. Printed 315 explicitly asks whether regularity alone together with a countable base suffices; this paper does not resolve that weaker assumption. Countable dense subsets alone cannot replace a countable base for arbitrary topological spaces, as its printed 310 footnote notes. Hausdorff separation used in printed 312 is retained; modern non-Hausdorff spaces are not admitted. Continuous closed-set separation is an inherited earlier Urysohn lemma reproved here, not a separately duplicated achievement. The compact-space corollary uses the historical Hausdorff setting and is not extended to arbitrary modern non-Hausdorff compact spaces. Original byline remains Urysohn, while Alexandroff's explicit posthumous scientific redaction is retained in the narrative credit rather than silently added as an original coauthor.",
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      "statement": "# Urysohn: turning a normal countable-base topology into a metric\n\nThe1925 article proves that a normal Hausdorff space with a countable base can be described by a compatible distance, using continuous separating functions. It distinguishes this from the weaker regularity condition left as a question in the paper.\n\nA topology tells us which sets count as open; a metric additionally assigns distances. This result gives a practical structural bridge: in a Hausdorff space with a countable base, the ability to separate disjoint closed sets by disjoint open neighborhoods is necessary and sufficient for a compatible metric. The paper builds continuous separators, arranges a countable family of them, and combines their differences into a distance with the same open sets.\n\n## Scientific scope\n\nThe exact 1925 target is the normal, second-countable Hausdorff-space criterion in printed 310 Hauptsatz and its countable-separator metric construction, not an unqualified attribution of the later familiar regular-second-countable theorem. Printed 315 explicitly asks whether regularity alone together with a countable base suffices; this paper does not resolve that weaker assumption. Countable dense subsets alone cannot replace a countable base for arbitrary topological spaces, as its printed 310 footnote notes. Hausdorff separation used in printed 312 is retained; modern non-Hausdorff spaces are not admitted. Continuous closed-set separation is an inherited earlier Urysohn lemma reproved here, not a separately duplicated achievement. The compact-space corollary uses the historical Hausdorff setting and is not extended to arbitrary modern non-Hausdorff compact spaces. Original byline remains Urysohn, while Alexandroff's explicit posthumous scientific redaction is retained in the narrative credit rather than silently added as an original coauthor.\n\n## People and systems\n\nPaul Urysohn\n\n\n\n## Original sources\n\n- [Paul Urysohn, Zum Metrisationsproblem, Mathematische Annalen94(1925),309–315: original-page GDZ OCR](https://gdz.sub.uni-goettingen.de/id/PPN235181684_0094)\n- [EuDML bibliographic record for Zum Metrisationsproblem](https://eudml.org/doc/159113)\n",
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