Let $\omega_{cf}$ be the countable space $\omega=\{0,1,2,3,\dots\}$ endowed with the cofinite topology $$\tau_{cf}=\{\emptyset\}\cup\{U\subseteq\omega:\omega\setminus U\mbox{ is finite}\}.$$ It is easy to see that the space $\omega_{cf}$ is second-countable, compact and connected, and so is its countable power $\omega_{cf}^\omega$.
Being motivated by the 5th problem of Kihara, I became interested in studying properties of metrizable subspaces of $\omega_{cf}^\omega$. Since $\omega_{cf}$ contains a discrete doubleton $\{0,1\}$, the countable power $\omega_{cf}^\omega$ contains the Cantor cube $\{0,1\}^\omega$ and consequently, contains a topological copy of any zero-dimensional metrizable separable space. What about higher-dimensional metrizable subspaces of $\omega_{cf}^\omega$?
Problem. Is each metrizable (or better Hausdorff) subspace of $\omega_{cf}^\omega$
- hereditarily disconnected?
- totally disconnected?
- zero-dimensional?
Let us recall that a topological space $X$ is
$\bullet$ zero-dimensional if $X$ has a base of the topology consisting of clopen subsets;
$\bullet$ totally disconnected if for any distinct points $x,y\in X$ there exists a clopen set $U\subseteq X$ such that $x\in U$ and $y\notin U$;
$\bullet$ hereditarily disconnected if each nonempty connected subspace of $X$ is a singleton;
$\bullet$ punctiform if each nonempty compact connected subspace of $X$ is a singleton.
Remark. By known Sierpinski Theorem of 1918, a compact connected Hausdorff space cannot be written as the countable union of pairwise disjoint nonempty closed sets. This implies that each compact metrizable subspace of $\omega_{cf}^\omega$ is a singleton, and each Hausdorff subspace of $\omega_{cf}^\omega$ is punctiform. In his comment to this MO-post, @Arno claims that he knows how to embed the Erdős space into $\omega_{cf}^\omega$, which would imply that the answer to the last subquestion (on zero-dimensionality of subspaces of $\omega_{cf}^\omega$) has negative answer.
