The quotient that identifies points indistinguishable by a real function algebra

Let $X$ be a compact Hausdorff space and let $A\subseteq C(X,\mathbb{R})$ be a real function algebra in the sense of Unital, point-separating, and nowhere-vanishing real function algebras on a compact Hausdorff space. Define a relation on $X$ by $x{\sim }_{A}y⟺f(x)=f(y)\text{ for every }f\in A\mathrm{.}$

This is an equivalence relation: equality gives reflexivity, symmetry of equality gives symmetry, and transitivity follows by applying transitivity of equality to $f(x)=f(y)$ and $f(y)=f(z)$ for each $f\in A$. The indistinguishability quotient of $X$ by $A$ is $Y_A:=X\mathrm{/}{\sim }_A,$ equipped with the quotient topology of the canonical surjection $q_A:X⟶Y_A,q_A(x)=[x{]}_A,$ as defined in The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection. Thus two points have the same image under $q_A$ exactly when no member of $A$ distinguishes them.