Base zero-dimensional spaces

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 $$s\in 2^{n+1}=\{0,1\}^{n+1}$$.

• @YCor, $[0,1]^{\omega}$ is usually called the Hilbert cube. – Mateusz Wasilewski Nov 25 '18 at 7:50
• ah OK. I had never read "Cantor cube" anyway. – YCor Nov 25 '18 at 8:36
• What if you take a subset of $2^\omega$, meeting each $B_s$ in a subset of cardinal $\aleph_1$? – YCor Nov 25 '18 at 23:41
• @YCor It seems that base zero-dimemsional sets are related to strong measure zero sets, so situtation becames to be interesting. – Taras Banakh Nov 26 '18 at 10:11
• I think the name of the tag is a mistake. People working in the area call cardinals like the ones you investigate cardinal characteristics. The name involves a field, techniques and a body of related results. Talk of "small uncountable cardinals" doesn't do that (seems like noise to me). – Andrés E. Caicedo Mar 13 '19 at 20:16

So, $$\mathfrak z=\mathrm{cov}(\mathcal M)$$ by a result of Fremlin and Miller.