I can understand (somewhat) why one would want to study non-Hausdorff topologies, since for example the Zariski topology is so important to algebraists, and the weak topology generated by lower semicontinuous functions on the real line also isn't T2.
But I have never heard of any non-T1 (not Kolmogorov, points may not be closed) space which is useful.
Moreover, for any non-T1 space isn't there a quotient map to a T1 space, just using the equivalence relation "two points are equivalent if they are topologically equivalent, ie share the same neighborhood system"? (This might even be a homeomorphism?)
Why isn't T1 the fourth topological axiom? In fact, I believe I saw one author use it as such. In any case I don't see any motivation to consider spaces which don't satisfy the axiom. Am I correct in this?
EDIT: Uh-oh. I thought that Kolmogorov/T0 was the same thing as T1 (i.e. I forgot that there exists something in between Hausdorff and "all points topologically indistinguishable").
Should I vote to close this question since it was based on a false premise?
I am interested in the reasons for studying T0 and not T1 spaces, so on one hand I do want to leave it open.
On the other hand, what I was really confused about is why we shouldn't reduce all non-T0 spaces to T0 spaces using the above mentioned equivalence relation, and thus why anyone would study non-T0 spaces?
I thought that the condition that all points are topologically distinguishable was only strictly weaker than Hausdorff, but equivalent to "all points are closed" -- I now realize my mistake.