## Largest Hausdorff quotient

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?

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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.

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(One can try to mod out $X$ by the relation "$x$ and $y$ cannot be separeted by disjoint open sets" —or its transtive closure, really— but the result is not Hausdorff; you can iterate this transfinitely, though, and you do get the space largest quotient. This is much more complicated/annoying/long to carry out) – Mariano Suárez-Alvarez Oct 15 2011 at 1:47
There are some set-theoretic issues here, though they can be dealt with. (You are currently quantifying over all compact Hausdorff spaces, which do not form a set.) – Daniel Litt Oct 15 2011 at 2:59
@Daniel, that is why I restricted to surjective maps. (In any case, the relation is well-defined even if one considers all maps and all $T_2$ sets, and one can easily show that there is a least topology satisfying the condition. One can quantify over all spaces!) – Mariano Suárez-Alvarez Oct 15 2011 at 3:03
Thanks, Marino! I was thinking in the way you describe in your second post and got stuck once I saw that the result is not Hausdorff. :) – mbasic Oct 15 2011 at 14:32

2) Another construction of the left adjoint to (Haus) -> (Top) works as follows: Let $X$ be a topological space, and consider the equivalence relation $\sim$ generated by: If $x,y$ cannot be separated by disjoint open sets, then $x \sim y$. Then $H(X):=X / \sim$ has the property that every map from $X$ into a Hausdorff space uniquely factors through $X \to H(X)$. If $H(X)$ was Hausdorff, we would be done. But this is not always the case. Instead, we have to repeat this construction: $X \to H(X) \to H(H(X)) \to H(H(H(X))) \to \dotsc$, then take the colimit $H^{\omega}(X)$, and make again $H^{\omega}(X) \to H(H^{\omega}(X)) \to \dotsc ...$. You can continue this for every ordinal number. Since $X$ is a set and all these maps are quotient maps, at some stage we get an isomorphism and then the desired Hausdorff quotient.