Two questions about implications between $\mathsf{DC}, \mathsf{BPI}$ and $\mathsf{AC}_\omega$

Question

Does the implication $\mathsf{DC} \implies \mathsf{BPI}$ hold?

And does the implication $\mathsf{BPI} \implies \mathsf{AC}_\omega$ hold?

I checked with Howard/Rubin's "Consequences of the Axiom of Choice" in part V where they list relations between forms, in particular, I could not find any mention of either of the above implications on page 324 and 326 respectively.

Answer 1

No implication whatsoever.

In Cohen's first model $\sf BPI$ holds, while both $\sf AC_\omega$ and $\sf DC$ fail.

In Shelah's model where every set has the Baire property, $\sf BPI$ fails, and $\sf DC$ holds (and consequently $\sf AC_\omega$ holds). One can also use for this Solovay's model or models of $\sf AD+\it V=L(\Bbb R)$, if one is willing to take large cardinals for granted.

Answer 2

Both implications are false, as can be verified at the Consequences website using the "list of all models with specified characteristics" feature (DC is form 43, BPI is 14, AC$_\omega$ is 8).