Dense-in-itself open sets in a subspace of the real line

Question

Given an uncountable set $X\subset [0,1]$ it is easy to write $X$ as a disjoint union of a perfect set $P$ (perfect in the subspace $X$) and an at most countable set $C$: just take $P$ as the set of condensation points of $X$ and $C$ as its complement in $X$. (we consider $X$ equipped with the subspace topology.)

I want to find an open set in $X$ which is dense-in-itself.

Is this possible? An open set of a dense-in-itself/perfect space is itself dense-in-itself, but from this point onwards I am most unsure how to proceed correctly.

Any comments will be appreciated. Thank you for your help.

Answer 1

It seems the following.

I want to find an open set in $X$ which is dense-in-itself. Is this possible?

It is not always possible. Let $\{a_n\}$ be an enumeration of all rational points of the unit segment $[0,1]$. Let $$X=\{(1/n, a_n):n\in\Bbb N\}\cup (\{0\}\times ([0,1]\setminus\Bbb Q)) \subset\Bbb R^2.$$ Then $X$ is an uncountable zero-dimensional metric space with a dense set of isolated points. But $X$ belongs to $\Bbb R^2$, not to $\Bbb R$. To solve this little problem we recall that each zero-dimensional second countable space can be embedded into Cantor set, which can be embedded into the unit segment.

(cited from “General Topology” by Ryszard Engelking. )

Answer 2

This is an easier version of Alex Ravsky’s idea.

Let $C$ be the usual middle-thirds Cantor set, let $D$ be the set of midpoints of the open intervals deleted in the construction of $C$, and let $X=C\cup D$; then $D$ is a dense set of isolated points in $X$, so $X$ has no dense-in-itself open subset.