The converse from Urysohn's lemma to DMC is open

As of 15 September 2026, $\mathrm{Z}\mathrm{F}$ proves that DMC implies Urysohn's lemma, while whether Urysohn's lemma implies DMC remains open. The countable-choice and BPI countermodels refute both Urysohn's lemma and the bounded Tietze extension conclusion; by contraposition of the proved DMC-to-Urysohn implication, they also fail DMC. Thus they realise $\mathrm{\neg }\mathrm{U}\mathrm{R}\mathrm{Y}\wedge \mathrm{\neg }\mathrm{D}\mathrm{M}\mathrm{C}$, which does not decide the converse $\mathrm{U}\mathrm{R}\mathrm{Y}\Rightarrow \mathrm{D}\mathrm{M}\mathrm{C}$.

• The positive implication. DMC implies Urysohn's lemma proves that DMC yields a continuous separation of any two disjoint closed sets of a normal space, using only finite menus of dyadic nodes. That theorem is the source of every "DMC suffices" clause on this page.

• The two countermodels. Relative consistency of Countable Choice without Urysohn's lemma and Relative consistency of BPI without Urysohn's lemma give, relative to $\mathrm{Con}(\mathrm{Z}\mathrm{F})$, models with countable choice, respectively BPI, in which Urysohn's lemma fails; the same models refute bounded Tietze extension for the same space by Brunner's endpoint obstruction also refutes bounded Tietze extension. The corresponding nonprovability conclusion for ZF is recorded as If ZF is consistent, ZF does not prove Urysohn's lemma, conditional on $\mathrm{Con}(\mathrm{Z}\mathrm{F})$.

• Why these do not settle the converse. Each countermodel fails Urysohn's lemma, so DMC implies Urysohn's lemma directly gives failure of DMC in that same model by contraposition. These are therefore models of $\mathrm{\neg }\mathrm{U}\mathrm{R}\mathrm{Y}\wedge \mathrm{\neg }\mathrm{D}\mathrm{M}\mathrm{C}$, not models of the conjunction needed to refute the converse. A model of $\mathrm{Z}\mathrm{F}+\mathrm{U}\mathrm{R}\mathrm{Y}+\mathrm{\neg }\mathrm{D}\mathrm{M}\mathrm{C}$ would settle the converse negatively, and no such model is known to the cited line of work.

• Dating. The status is dated because it is a report about the present state of the subject rather than a mathematical theorem; if the question is answered, this item must be replaced by the corresponding theorem and its proof, not reworded.