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.