(This is based on my earlier question, but I think this one would be easier to answer.)


Let $\langle X,\mathbf{\delta} \hspace{.01 in} \rangle$ be a separated proximity space, and let $\cal{T}\hspace{.04 in}$ be the induced topology on $X$.
Then, ZF proves that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is regular Hausdorff, and
ZF + (Dependent Choice) $\;$proves that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is completely regular.

Does ZF prove that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is completely regular?

(This is based on my earlier question, but I think this one would be easier to answer.)


Let $\langle X,\mathbf{\delta} \hspace{.01 in} \rangle$ be a separated proximity space, and let $\cal{T}\hspace{.04 in}$ be the induced topology on $X$.
Then, ZF proves that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is regular Hausdorff, and
ZF + (Dependent Choice) $\;$proves that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is completely regular.

Does ZF prove that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is completely regular?