The question was very popular over on MSE but seems to have left everyone speechless. Maybe someone here can help?

Definition: Suppose $X$ is a compact connected Hausdorff space and $D \subset X$ countable and dense. We say $D$ is divisible to mean for every open $U \subset X$ there exists a partition $D \cap U = D_1 \cup D_2$ with $D_1$ and $D_2$ both dense in $U$. We call $X$ divisible to mean it is separable and each countable dense subset is divisible.

Is every compact connected Hausdorff space divisible?

The property is easy to show if $X$ is metric rather than just Hausdorff. Just let $U_1,U_2, \ldots $ be a countable basis. Since $X$ is connected each $U_n$ contains a proper subcontinuum which surjects onto $[0,1]$ by Urysohn's lemma. Hence each open subset is uncountable. So choose distinct $a_1,b_1 \in U_1$ and proceed by induction. At each stage we have selected only finitely many $a_n,b_n$ and $U_m$ contains infinitely many other elements. So select distinct $a_m,b_m \in U_m - \{a_1,b_2, \ldots , a_{m-1},b_{m-1}\}$ and define $D_1 = \{a_1,a_2,\ldots\}$ and $D_2 = \{b_1,b_2,\ldots\} \cup (D - D_1)$. Then $D_1 \cap U$ and $D_2 \cap U$ form the desired partition for any choice of $U$. So $X$ might be called simultaneously divisible.

The property fails for (disconnected) spaces with an isolated point. Because if $\{x\} \subset X$ is open $x$ must be an element of exactly one of $D_1,D_2$ which means the other is not dense. I am not sure if non-divisible spaces are any easier to come by if we only perfect spaces instead of connected spaces.

Under AC being Hausdorff is necessary. For the cofinite topology is compact and connected and any infinite subset is dense. Without AC $D$ might be an amorphous set.

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    $\begingroup$ What you are calling "divisible" is related to resolvability. There is a closely related question here (different, but related): mathoverflow.net/questions/275424/…. $\endgroup$ – Will Brian Apr 17 '18 at 13:03
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    $\begingroup$ $[0, 1]^{\mathfrak{c}}$ has a countable dense submaximal subspace - See Section 4 in I. Juhasz, L. Soukup, Z. Szentmiklossy, D-forced spaces: A new approach to resolvability, Topology and its applications, Volume 154, Issue 16, August 2007, Pages 2997-3004 $\endgroup$ – Ashutosh Apr 17 '18 at 14:05
  • $\begingroup$ @Ashutosh: It's Volume 153 (2006), Issue 11, Pages, 1800–1824. $\endgroup$ – user87690 Apr 17 '18 at 15:28
  • $\begingroup$ @user87690 Thanks for the correction - The incorrect reference was for Realcompactness in maximal and submaximal spaces by Hernandez-Hernandez et al. which offers another proof of this in Theorem 3. $\endgroup$ – Ashutosh Apr 17 '18 at 15:38
  • $\begingroup$ Suddenly you mention separable. Strange. Can you say more? $\endgroup$ – Wlod AA Apr 18 '18 at 5:20

Here are some details following [1].

Claim: There is a countable dense $X \subseteq [0, 1]^{\mathfrak{c}}$ which does not have a dense co-dense subset.

Proof: Follows from (1) + (2) below.

(1) Every countable dense subspace of $2^{\mathfrak{c}}$ is homeomorphic to a countable dense subspace of $[0, 1]^{\mathfrak{c}}$.

Proof of (1): Use the fact that for every countable dense $D \subseteq 2^{\omega}$, $2^{\omega} \setminus D$ is homeomorphic to the Baire space $([0, 1] \setminus \mathbb{Q})^{\omega}$.

(2) $2^{\mathfrak{c}}$ has a countable dense subspace $X$ which has no dense codense subset.

Proof of (2): (Alas et al. [1]) Let $\{A_i : i < \mathfrak{c}\}$ list all infinite coinfinite subsets of $\omega$. Inductively, try to construct $X_i = \{x_{i, n}: n < \omega\} \subseteq 2^{\mathfrak{c} +i}$ for $i < \mathfrak{c}$ such that the following hold.

(a) $X_i$ is dense in $2^{\mathfrak{c}+i}$.

(b) If $i < j$, then $x_{j, n} \upharpoonright (\mathfrak{c} + i) = x_{i, n}$.

(c) If $\{x_{i, n}: n \in A_i\}$ is dense codense in $X_i$, then $\{x_{i+1, n}: n \in A_i\}$ is open in $X_{i+1}$.

There is no problem at stages $i = 0$, limit.

At stage $i + 1$: If $X_i$ has no dense codense subset, we terminate the construction and put $X = X_i$ - So $X \subseteq 2^{\mathfrak{c} + i} \cong 2^{\mathfrak{c}}$ and we are done. Otherwise, choose an infinite cofinite $A \subseteq \omega$ such that

(i) $\{x_{i, n}: n \in A\}$ is dense codense in $X_i$ and

(ii) if $\{x_{i, n}: n \in A_i\}$ is dense codense in $X_i$, then $A = A_i$

and define $x_{i+1, n} = x_{i, n} \cup \{(i, 1)\}$ if $n \in A$ and $x_{i+1, n} = x_{i, n} \cup \{(i, 0)\}$ if $n \notin A$. As noted above we can assume that the construction can be carried out at every $i < \mathfrak{c}$ and we set $X = \{\bigcup_{i < \mathfrak{c}} x_{i, n} : n < \omega\}$. It is easily checked that $X$ is dense in $2^{\mathfrak{c} + \mathfrak{c}} \cong 2^{\mathfrak{c}}$ and has no dense codense subset.

[1]: Alas et al., Irresolvable and submaximal spaces: Homogeneity versus σ-discreteness and new ZFC examples, Topology and its Applications, Volume 107, Issue 3, 4 November 2000, Pages 259-273

  • $\begingroup$ Thank you. I had seen reference to irresolvable sets in passing but I think they were mostly formulated in terms of the "every dense set is open" property which I believe is stronger. I hadn't thought it was an area with so much recent activity! $\endgroup$ – Daron Apr 18 '18 at 10:49

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