The inclusion of the full subcategory of Hausdorff topological spaces into the category of topological spaces has a left adjoint, which can be proven easily by the Adjoint Functor Theorem (see for example, S. MacLane, Categories for Working Mathematicians). To every topological space this left adjoint associates a Hausdorff space called the largest Hausdorff quotient.

Do you know a reference in which this left adjoint is constructed explicitly?