Why haven't I heard of the Hausdorff quotient? I just today discovered that, similar to the Kolmogorov quotient, there is a universal quotient from any topological space into a Hausdorff space, which any continuous map into a Hausdorff space factors through. This seems to be all but invisible on the internet: the only  things I could find talking about it are a single MathOverflow post and a Bachelors dissertation citing it. This hasn't been mentioned in any topology material I have seen, and I only thought to look because I stumbled across it myself. The Kolmogorov quotient is mentioned everywhere, and seen as very important, so why is the Hausdorff quotient so neglected?
Similarly, there is (I am near certain, as the same proof works for each of them) a universal $T_1$ quotient, which I also haven't seen mentioned, and doesn't seem to have a single page discussing it.
The only reason I can think that this might be is that the quotient may be quite trivial for most spaces that crop up, as it seems more 'aggressive' than the Kolmogorov quotient in combining points. I'm not aware of any non-Hausdorff connected spaces for which the quotient is not the topological space of size $1$, for example. On the other hand, I only discovered this today. And it seems unlikely that all quotients are that trivial.
I would also be interested to know any interesting examples of the $T_1$ and Hausdorff quotient that you're aware of.
 A: Captain Lama’s answer is excellent, so I shan’t repeat its points, but I’ll add a few more illustrative examples.
The difference between $T_0$ and Hausdorff is often visible in the specialization (pre-)order on a space $X$: $x \leq_X y$ if every open neighbourhood of $x$ contains $y$, or equivalently, $x$ is in the closure of $\{y\}$.  Then $X$ is $T_0$ just if this is a partial order ($\leq_X$-equivalent points are equal), and $T_1$ just if it is a discrete order (any distinct elements are incomparable); so in particular it’s discrete for a Hausdorff space.  Any continuous map preserves this order, so the Hausdorff quotient must at least identify points that are specialisation-comparable.
In particular, many important examples of non-Hausdorff spaces have connected specialisation orders (at least within their topological components do) — for instance, the Zariski topology on the spectrum of a ring.  In that example and many others, there is often (at least if the ring is an integral domain) a generic point, whose closure is the whole space, and so which is $\leq$-maximal; so the Hausdorff quotient identifies all points.
On the other hand, the specialisation order can help us find a connected non-Hausdorff space with non-trivial Hausdorff quotient: Take your favourite Hausdorff space, and add a new point specialising some existing point.  E.g. Take $X := \mathbb{R} \cup \{0'\}$, with a set $U \subseteq X$ open just if its intersection with $\mathbb{R}$ is open and additionally if $0' \in U$ then also $0 \in U$.  Then its Hausdorff quotient just identifies $0'$ with $0$, and is isomorphic to $\mathbb{R}$.
A: The Hausdorff quotient is not that exotic. It is known under the name Hausdorffization or Hausdorff reflection. Here are some links:

*

*The Topology Wiki. See https://topospaces.subwiki.org/wiki/Hausdorffization.


*nLab. See https://ncatlab.org/nlab/show/Hausdorff+space#HausdorffReflections.


*In this forum: https://math.stackexchange.com/search?q=Hausdorffication+.
Whether it is an important concept (or if it is a less important concept than the Kolmogorov quotient) is a philosophical question. The answer depends on your personal interests.
The Hausdorff quotient of a space $X$ is always a quotient of the Kolmogorov quotient of $X$, so it is indeed "more aggressive" in identifying points of $X$. You write

I'm not aware of any non-Hausdorff connected spaces for which the quotient is not the topological space of size $1$.

Peter LeFanu Lumsdaine has given a non-trivial example in his answer. For $X = \mathbb R$ we obtain the line with two origins (see e.g. here). This is a famous example for a non-Hausdorff manifold.
A: Personally, I would say that this is because it's a construction that is sort of "in-between worlds", and is not super relevant to any of those worlds. Precisely, by definition this Hausdorff quotient is relevant exactly to people who work with non-separated spaces, but who care about the Hausdorff condition. And I would say that this population is very slim.
Usually, either you only work with nice spaces, including metric spaces, topological manifolds, CW-complexes, that sort of things, and in that case obviously you don't need to make your spaces separated (they already are).
Or you work with more general spaces, and in that case it usually means that you just don't really care about the Hausdorff condition. It's of course always nice if it's satisfied, but it doesn't have to play a special role. This is in contrast with the Kolmogoroff condition, which is often crucial even for people who work with very general spaces, because if a space is not even $(T_0)$ then its topology does not "see" all the points in the space, which sort of means you are not working with the "correct" set (so you take the appropriate quotient).
So in general I would say that for all those "universal" constructions (usually reflection functors for a reflective subcategory) which "give property $A$ to an object in a universal fashion", the question is: "do people who work with non-$A$ objects still care about property $A$ ?". For instance, people who work with non-abelian groups usually still care about the abelian condition. People who work with non-compact or non-Kolomogoroff spaces still care about compact/Kolmogoroff spaces. So the abelianization, compactification, and Kolmologoroffication (?) functors are considered interesting. But that is much less the case with Hausdorff spaces (at least this is my opinion on the matter !).
(By the way the fact that as you noted in most natural cases when you try to make your space Hausdorff it just becomes discrete tends to show that the Hausdorff condition is kind of an "all-or-nothing" property in practice, which is in line with what I argued.)
