At the beginning of the book Tame topology and o-minimal structures, by Lou van den Dries, it said that all the open sets of $\mathbb{R}^n$ can be obtained from a single polynomial equation. More precisely, there are $p,q \in \mathbb{N}$ and a polynomial $P(x_0,x_1,\ldots,x_n,y_1,\ldots,y_p,z_1,\ldots,z_q)$ with coefficients in $\mathbb{Z}$ such that for any open subset $U \subseteq \mathbb{R}^n$, there is a real $r \in \mathbb{R}$, such that $$ U = \{(a_1,\ldots,a_n) \in \mathbb{R}^n \mid \exists b_1,\ldots,b_p \in \mathbb{R}, \exists c_1,\ldots,c_q \in \mathbb{Z} \text{ such that } P(r,a_1,\ldots,a_n,b_1,\ldots,b_p,c_1,\ldots,c_q) = 0 \}. $$ Even the case $n = 1$ seems hard. In the book, it is said that "those familiar with descriptive set theory and with Matijasevich's theorem on diophantine sets may find it an amusing exercise" to prove this fact. I admit that I'm not familiar with Matijasevich's theorem, and that I don't know a lot about descriptive set theory, but I'm still curious to see a proof. I would also like to know if we can describe explicitely this $P$.
This fact can be used as a quick way to show that in the structure $(\mathbb{R},+,\cdot,\mathbb{N})$ we can define all open sets, and from this it is not too hard to prove that all projective sets are definable.