= Solution
{c}
Take distinct classes $p(x)\ne p(y)$. The failure of $x\mathcal Ry$ supplies a Hausdorff space $Y$ and a continuous map $g:X\to Y$ with $g(x)\ne g(y)$. Choose disjoint open neighborhoods $V,W$ of these two images. Their inverse images $A=g^{-1}(V)$ and $B=g^{-1}(W)$ are disjoint open subsets of $X$, and part (ii) says both are saturated.
The <quotient topology> therefore makes $p(A)$ and $p(B)$ open, since their inverse images are precisely $A,B$. They are disjoint as well: an equivalence class meeting both would, by saturation, be contained in both. They contain $p(x)$ and $p(y)$ respectively. Thus \b[$X/\mathcal R$ is Hausdorff]. This construction is the <Hausdorff reflection by continuous maps>.
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