I consider a set $X\subset\mathbb{R}^d$ and an homeomorphism $g : X\to Y\subset\mathbb{R}^d$. I woul like to prove that if $Y$ has non empty interior (for the topology of $\mathbb{R}^d$) then $X$ also.
My attempt was the following : let $U$ ben an open set of $\mathbb{R}^d$ such that $U\subset Y$, we have that $g^{-1}(U)$ is open in $X$ since $g$ is an homeomorphism, however I do not see why it should be an open set in $\mathbb{R}^d$.
If you have counter example to propose I would be glad to see it please.