Denote by $S$ the set of closed points in $X=\mathrm{Spec}\:\mathbb{R}[x_\alpha]$ ($\alpha \in \mathbb{Q}$) that have $\mathbb{R}$ as their residue field. There is an injective map from the set of continuous functions $\mathbb{R}\to\mathbb{R}$ to $S$. Is there a locally closed subset $Y\subset X$ such that $Y\cap S$ is equal to the image of this map?