= Hausdorff reflection by continuous maps
{c}
{title2=$x\sim y\iff f(x)=f(y)\text{ for every continuous map to a Hausdorff space}$}
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.
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