What is recorded here?
The1925 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. A 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.
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Scientific scope
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.
Read the original work
- Paul Urysohn, Zum Metrisationsproblem, Mathematische Annalen94(1925),309–315: original-page GDZ OCR
- EuDML bibliographic record for Zum Metrisationsproblem
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People and systems
Paul Urysohn
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English catalogue narrative
Urysohn: turning a normal countable-base topology into a metric
The1925 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.
A 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.
Scientific scope
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.
People and systems
Paul Urysohn