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? 



Largest Hausdorff quotient
Consider the equivalence relation $\sim$ on your space $X$ such that $x\sim y$ iff $x$ and $y$ have the same image under all surjective continuous maps $f:X\to Y$ with codomain $Y$ a Hausdorff space. Put on the set $X/\sim$ the least topology which makes all those maps continuous, and you have the space you want. I doubt there is any actual reference for this.


