On subcompactness and countable subcompactness of metrizable spaces in ZF

We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every countably subcompact. (ii) A $\mathbf{X}=(X,T)$ compact if only it relative to $T$. (iii) For $\mathbf{X}=(X,T)$, the following are equivalent: \noindent(a) $\mathbf{X}$ compact; \noindent(b) for open filter $\mathcal{F}$ of $\mathbf{X}$, $\bigcap \{\overline{F}\colon F\in \mathcal{F}\}\neq \emptyset $; \noindent(c) also show: (iv) The negation each statements, (a) (b) subcompact, (c) relatively consistent with ZF. (v) AC family $\{\mathbf{X}_{i}\colon i\in I\}$ spaces, $\{\mathcal{B}_{i}\colon such that $i\in I$, $\mathcal{B}_{i}$ a base $\mathbf{X}_{i}$, Tychonoff product $\mathbf{X}=\prod_{i\in I} \mathbf{X}_{i}$ respect standard $\mathcal{B}$ generated by I\}$.