For a proof that $\mathbb{R}^3\setminus \mathbb{Q}^3$ is simply connected using Baire category theorem I need to approximate an homotopy $H : [0,1]\times \mathbb{S}^1 \to \mathbb{R}^3$ from a loop $\gamma$ to a point $x$ by another avoiding a point $q$ ($\gamma(z) \neq q $ $\forall z\in\mathbb{S}^1$ and $x\neq q$) with an arbitraty error $\varepsilon$. In symbols : $H' : [0,1]\times \mathbb{S}^1 \to \mathbb{R}^3\setminus\{q\}$ such that $H'(0,z) = \gamma(z)$ , $H'(1,z) = x$ and $\underset{(t,z)\in [0,1]\times\mathbb{S}^1}{\sup}\vert H(t,z) - H'(t,z)\vert < \varepsilon$.

I do not even know if it is true.

Also, if you know another proof of the simple connectess of $\mathbb{R}^3\setminus\mathbb{Q}^3$ (if it is simply connected) it would be nice to be able to see it.