I have read that given any topological space $X$ you can construct a Hausdorff space $h(X)$ as a quotient of $X$ which is universal with respect to maps from $X$ to a Hausdorff space. This means there is a quotient map $q:X\to h(X)$ and if $f:X\to W$ is a map to a Hausdorff space, $W$, then there is a unique map $g:h(X)\to W$ such that $gq=f$.
It seems you can construct the equivalence relation $\sim$ on $X$ so that $X/\sim$ by saying $x\sim y$ $\Leftrightarrow$ $f(x)=f(y)$ for every map $f:X\to W$ to Hausdorff $W$; this is essentially an application of a general adjoint theorem. I would like to know if it is possible to construct $h(X)$ using a transfinite process of taking quotients of $X$.
It seems like the following should be a start:
Let $X_0=X$. Say $x\sim_{0} y$ if $x$ and $y$ can't be separated by disjoint open sets in $X$. Let \sim_{0}' be the transitive closure of this relation so you now have an equivalence relation. Now let X_1=X_0/\sim_{0}'.
Since it seems $X_1$ is not necessarily Hausdorff (I am not even sure it is $T_1$) you can construct the equivalence relation \sim_{1}' on $X_1$ and a quotient map X\to X_2=X_{1}/\sim_{1}' by replacing $0,1$ with $1,2$. You should be able to continue this construction by transfinite induction to get a quotient $X_{\alpha}$ of $X$ for each ordinal $\alpha$. For instance, if $\alpha$ is a limit ordinal, define x\sim_{\alpha}'y $\Leftrightarrow$ there is a $\lambda<\alpha$ such that x\sim_{\lambda}'y.
Does this transfinite sequence of spaces stabilize to $h(X)$? If so, how does one show this?
If this initial approach is off the mark, is there a more appropriate inductive approach which does stabilize to $h(X)$?