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