Hausdorff reflection by continuous maps ($x\sim y⟺f(x)=f(y)\text{ for every continuous map to a Hausdorff space}$)

For a topological space $X$, the displayed relation is an equivalence relation. Give its quotient the quotient topology. Every inverse image under a continuous map from $X$ to a Hausdorff space is a saturation of a subset. Distinct quotient classes can be distinguished by some such map; disjoint open neighborhoods of their images pull back to disjoint saturated open sets, whose quotient images are open. Thus the quotient is Hausdorff. Each continuous map from $X$ to a Hausdorff space factors uniquely through this quotient, and the factor is continuous by the quotient topology. This is its universal reflection property.