TITLE: nonhausdorff dimension
QUESTION [12 upvotes]: if $X$ is a topological space, a first step in making $X$ hausdorff is taking the quotient $H(X)=X/\sim$, where $\sim$ is the equivalence relation generated by: if $x,y$ cannot be seperated by disjoint open sets, then $x \sim y$. observe that $X$ is hausdorff, when $X \to H(X)$ is an isomorphism, and that for every hausdorff space $K$ the map $Hom(H(X),K) \to Hom(X,K)$ induced by the projection $X \to H(X)$ is bijective.
by a fairly general categorical argument, we can construct from this the free functor from topological spaces to hausdorff spaces (i.e. it's left adjoint to the forgetful functor): for ordinal numbers $\alpha$, define the functor $H^\alpha$ (together with natural transformations $H^{\alpha} \to H^{\beta}, \alpha < \beta$) by $H^0 = id, H^{\alpha+1} = H \circ H^\alpha$ and $H^\alpha = colim_{\delta < \alpha} H^\delta$. for every topological space $X$ there is an ordinal number $\alpha$ such that $H^\alpha(X) = H^{\alpha+1}(X)$, then $H^\alpha(X)$ is the free hausdorff space associated to $X$. define $h(X)$, the "nonhausdorff dimension" to be the smallest such ordinal number $\alpha$. every ordinal number arises as a nonhausdorff dimension(!).
I've came up with this with a friend and we don't know of any literature about it. perhaps someone of you has already seen it elsewhere? there are some further questions: every $H^\alpha(X)$ is a quotient of $X$, but how can we describe the equivalence relation explicitely? what is the intuition for a space $X$ to have nonhausdorff dimension $\alpha$? are there known classes of topological spaces whose nonhausdorff dimension can be bounded? and of course: is there some use for the nonhausdorff dimension? ;-)

REPLY [5 votes]: An unpublished, but quite well distributed, reference is "Appendix A, Compactly Generated Spaces" to the 1978 Ph.D. thesis of L. Gaunce Lewis, Jr.  Proposition 4.1 discusses the existence of the left adjoint (Hausdorffification) by way of Freyd's adjoint functor theorem, and Construction 4.3 explains the transfinite induction by iterating $X \to JX$ where JX is essentially the quotient of X discussed by Martin Brandenburg.  (Lewis works with weak Hausdorff spaces, but the story is the same.)  A published version of something much like this appears in Lewis, May and Steinberger's book "Equivariant Stable Homotopy Theory" (Springer LNM 1213), where the spectrification functor L is constructed in Section 1
of the Appendix (pages 475-481) by this kind of transfinite induction.