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Jean-Baptiste Vienney (Oct 12 2024 at 17:02):This MO question asks for a characterization of topological spaces which are “Lawvere metrizable”.
The given answer is that topological spaces with a countable base are Lawvere metrizable. This is a rather unsatisfactory answer since this is a sufficient but not necessary condition.
Indeed, every discrete space is Lawvere metrizable (in fact metrizable) by defining if and . But if is not countable, then doesn’t have a countable base. (A base for necessarily contains each singleton).
Such an answer is not so surprising since the question was also asking for an analogue of the Urysohn metrization theorem which also gives a sufficient but not necessary condition.
Is anyone aware of a characterization of Lawvere metrizable topological spaces? (akin to a result such as the Nagata–Smirnov metrization theorem)
Jean-Baptiste Vienney (Oct 12 2024 at 17:42):Well, looking at both the answer, Urysohn’s theorem and Nagata-Smirnov’s theorem, I guess the characterization could be “a topological space is Lawvere metrizable iff it has a countably locally finite base”.
James E Hanson (Oct 15 2024 at 23:29):I wrote an answer giving an exact characterization in the literature along with some discussion of related results, although I think it's probably pretty unsatisfying. The exact characterization is morally pretty close to 'the topology is induced by a quasi-uniformity with a countable base.'
Also, having a countably locally finite base is not a necessary condition for being Lawvere-metrizable (although it and a weaker condition is sufficient).
Jean-Baptiste Vienney (Oct 16 2024 at 14:20):Thank you very much for your very complete answer!
Jean-Baptiste Vienney (Oct 16 2024 at 14:29):That’s exciting to see that the Sion, Zelmer paper giving the sufficient condition which is weaker than countably locally finite base is from 2018 (EDIT: in fact it is from 1967 :)). I’ll read about quasi-uniformities to get a better sense of the full story.
Jean-Baptiste Vienney (Oct 16 2024 at 14:30):It would be great if the sufficient condition of Sion, Zelmer could be weakened to a necessary and sufficient one.