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Algebraically cutting out Polynomial constraints on the values of continuous functions from all functions $\mathbb{R}\to\mathbb{R}$

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?

Algebraically cutting out continuous functions from all functions $\mathbb{R}\to\mathbb{R}$

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

Polynomial constraints on the values of continuous functions $\mathbb{R}\to\mathbb{R}$

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?

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user158636
user158636

Algebraically cutting out continuous functions from all functions $\mathbb{R}\to\mathbb{R}$

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