Can a countable dense subset be split into two disjoint dense subsets?
Solution 1:
Below is Ashutosh's answer from MathOverflow:
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