Homeomorphic to a product space

Question

Consider $\mathbb{R^2}$ with usual topology. We define $\mathcal{S}$ as the equivalence relationship on $\mathbb{R^2}$ that identifies to one point all elements in $\mathbb{Q^2}$. I need to check if $X:= \mathbb{R^2}/\mathcal{S}$ is homeomorphic to a product space. Of course, the vertical lines in $X$ are not all homeomorphic, but of course that doesn't prove anything, apart from the fact that I'll need to apply a homeomorphism different from the identity before identifying the space with a product. I don't know how to proceed, can you give me some advice?

Answer 1

Hint: For $A × B ⊆ X × Y$ we have $\overline{A × B} = \overline{A} × \overline{B}$. It follows that a point $(x, y) ∈ X × Y$ is closed if and only if $x$ is a closed point in $X$ and $y$ is a closed point in $Y$, and $(x, y)$ is a dense point in $X × Y$ if and only if $x$ is a dense point in $X$ and $y$ is a dense point in $Y$.