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. 



Question about product topology
In the following [paper](http://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03561-7/S0002-9939-96-03561-7.pdf), van Engelen constructs a strongly homogeneous, zero dimensional, Borel, dense subspace $Y$ of $2^{\mathbb{N}}$ such that while $Y$ does not admit a topological group structure, $Y^2$ does. I haven't read the proof but maybe you can use it to construct a counterexample in the context of $\mathbb{R}$?


