Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.

Is $S\times S$ homeomorphic to $S$?

By Lusin scheme, if $S$ is the set of rationals or irationals , I can see this statement is true.

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.

Is $S\times S$ homeomorphic to $S$?

By Luzin scheme, if $S$ is the set of rationals or irationals , I can see this statement is true.

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.

Is $S\times S$ homeomorphic to $S$?

I can see if $S$ is rationals or irationals , this statement is true.

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.

Is $S\times S$ homeomorphic to $S$?

By Lusin scheme, if $S$ is the set of rationals or irationals , I can see this statement is true.