Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal U$ of $X$ has a disjoint refinement $\mathcal V\subset\mathcal B$.
It can be shown that
(1) each countable regular space is base zero-dimensional;
(2) the Cantor set is not base zero-dimensional.
Let $\mathfrak z$ be the smallest cardinality $|Z|$ of a subset $Z\subset\mathbb R$, which is not base zero-dimensional. It follows that $\aleph_1\le\mathfrak z\le\mathfrak c$. So, $\mathfrak z$ is a typical small uncountable cardinal.
Problem 1. Is $\mathfrak z$ equal to some known small uncountable cardinal? Is $\mathfrak z=\mathfrak c$ under MA or PFA?
Edit 1 (written following a suggestion of @user64494): I found a (relatively) simple solution to my original question (about the base zero-dimensionality of the Cantor set) and then edited my post asking the next natural question in this context (about the base zero-dimensionality of uncountable sets of the real line).
By the way, a base $\mathcal B$ witnessing that the Cantor cube $2^\omega$ is not base zero-dimensional consists of the sets $$B_s:=\{x\in 2^\omega:x{\restriction}n=s\mbox{ and }(x(n)\ne s(n)\Rightarrow x(n{+}1)=0)\}$$where $n:=\{0,\dots,n-1\}\in\omega$ and $s\in 2^{n+1}=\{0,1\}^{n+1}$.
