as a continuation to my question here: Is cantor set homeomorphic to the unit interval?
I can't see how can it is be true that any uncountable $G_\delta$ set of $\mathbb{R}$, has a subset homeomorphic to $[0,1]$. Isn't the process described here, from John C. Oxtoby's book Measure and Category, similar to the construction of the Cantor set?
Thank you!
To give this an answer, expanding on Carl Mummert's comment:
It doesn't.
Lemma 5.1 of your text asserts that each uncountable $G_\delta$ contains a nowhere dense closed subset $C$ that can be mapped continuously onto $[0,1]$, i.e. there is a continuous surjective map $f : C \to [0,1]$. It is not asserted that $f$ is injective nor that it has a continuous inverse, nor that $C$ is homeomorphic to $[0,1]$ via some other map. In general it is not. For example, if $E$ is the Cantor set, by considering connectedness one can see that any continuous $g : [0,1] \to E$ is constant, so no surjective $f : C \to [0,1]$ can possibly have a continuous inverse.