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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 $\omega_{cf}^\omega$

  1. hereditarily disconnected?
  2. totally disconnected?
  3. 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.

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  • $\begingroup$ @bof The answer to your first comment is yes. To answer my first question, it suffices to prove that each nonempty connected metrizable subspace of $\omega_{cf}^\omega$ is a singleton. $\endgroup$ Jul 25, 2020 at 7:23
  • $\begingroup$ @bof Concerning your second comment, unfortunately, I am not an expert in false memory syndromes. Yet, I follow the standard terminology from the Engelking's book in which Engelkings writes (in Historical Notes to the section "Various kinds of diconnectedness") that hereditarily disconnected spaces were introduced by Hausdorff in 1914 who called them "totally diconnected spaces", but after Sierpinski (1921) whose introduced totally disconnected spaces (in their modern meaning) the terminology has changed. But this happened long-long ago (by the way, what is your age)? $\endgroup$ Jul 25, 2020 at 7:25
  • $\begingroup$ I too learned that totally disconnected meant that the connected components are singletons, while totally separated meant points can be separated by clopen sets. $\endgroup$ Jul 25, 2020 at 8:09
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    $\begingroup$ @shane.orourke This simply means that in different areas of mathematics, the same mathematical objects have different names (this is contrary to a definition of mathematics as a science that gives the same names to different concepts). In this situation, it is better to follow the standards from the main discipline to which a concept is attributed. In our case this is General Topology where Engelking's "General Topology" is a standard reference. $\endgroup$ Jul 25, 2020 at 9:07
  • $\begingroup$ @bof Wikipedia is a good reference (especially if it is written by a specialist). Unfortunately, in this case we have two rather standard references (Engelking and Willard) using the same terminology in two different senses. And this is bad. $\endgroup$ Jul 25, 2020 at 11:30

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