Choice ledger for Baire, Urysohn, Stone, and Tychonoff

Ledger: separable complete metric Baire is a theorem of $\mathrm{Z}\mathrm{F}$; complete metric Baire is DC; compact-Hausdorff Baire is exactly DMC; Baireness of products of compact Hausdorff spaces is DC; DMC implies Urysohn's lemma, while, relative to the consistency of ZF, countable choice and BPI are each consistent with the failure of Urysohn's lemma and of bounded Tietze extension; Stone follows from AC, while relative to the consistency of ZF both DC and BPI are separately consistent with a metrizable space having an open cover with no locally finite open refinement; the stronger per-cover effective refinement assertion for discrete metrizable spaces implies AC; compact Hausdorff products and cofinite products have the strength of BPI; compact $T_1$ and arbitrary compact products have the strength of AC. DC implies DMC; if ZF is consistent, ZF does not prove DMC; and DMC-to-DC over $\mathrm{Z}\mathrm{F}$ remains open.

• Baire rows. Separable complete metric spaces are Baire in ZF is ZF, whereas Dependent Choice is equivalent to the complete-metric Baire principle over ZF proves over ZF that the unrestricted complete-metric Baire principle is equivalent to DC; DMC makes every compact Hausdorff space Baire gives DMC implies compact-Hausdorff Baire, Compact Hausdorff Baire implies DMC gives the converse and Compact Hausdorff Baire is equivalent to DMC packages the equivalence; DC is equivalent to Baireness of compact-Hausdorff products is the product row, which is DC and not merely DMC.

• Urysohn rows. DMC implies Urysohn's lemma is the positive row; Relative consistency of Countable Choice without Urysohn's lemma and Relative consistency of BPI without Urysohn's lemma are the two separations, and Brunner's endpoint obstruction also refutes bounded Tietze extension records the bounded-Tietze consequence. The status of the converse is The converse from Urysohn's lemma to DMC is open.

• Stone rows. Stone's theorem, under choice: every metric space is paracompact is the AC row; Relative consistency of DC with failure of Stone's theorem and Relative consistency of BPI with failure of Stone's theorem are relative-consistency separations from DC and BPI, respectively. In the BPI model the sharper obstruction is a metrizable space with an open cover having no point-finite open refinement, hence no locally finite open refinement; Effective metacompactness for discrete metric spaces implies AC is the effective strengthening, and the exact strength of the ordinary theorem is recorded as open in The exact choice strength of Stone's theorem remains open.

• Tychonoff rows. Cofinite products and compact Hausdorff products have the strength of BPI (Products of cofinite spaces are compact exactly under BPI, and the compact Hausdorff equivalence cited there); compact $T_1$ products and arbitrary compact products have the strength of AC (The compact T1 product theorem is equivalent to AC, The arbitrary compact product theorem is equivalent to AC).

• Principle rows. DC implies DMC and, assuming the consistency of ZF, DMC is not a ZF theorem (If ZF is consistent, DMC is not provable in ZF); BPI does not imply DMC (BPI does not imply DMC); the qualifications over $\mathrm{Z}\mathrm{F}\mathrm{A}$ and the openness of the reversal are in DMC, Multiple Choice, and AC qualifications and DMC versus DC over ZF remains open. No strict DMC-versus-DC claim over $\mathrm{Z}\mathrm{F}$ is made anywhere in this ledger.