(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same open sets around them.)
For a space that is not $T_0$, we can uniquely form a $T_0$ space from it by taking the Kolmogorov quotient, which just means sending two points to the same point in the quotient iff they are, so to speak, a counterexample to $T_0$-ness. This induces an isomorphism (i.e. order-preserving, union-/finite intersection-compatible bijection) on the topologies of the old space and the new space
, and the new space is homeomorphic to any other $T_0$ space with the same topology.
Wikipedia seems to say that we can toggle $T_0$-ness on and off and get analogous theorems in many cases. But it also says, "[...] it may also be easier to allow structures that aren't T0 to get a fuller picture."
In short, I'm wondering:
- What does point-set topology lose if we require topological spaces to always be $T_0$? (Probably nothing, in pure point-set topology.)
- What non-$T_0$ spaces in other fields have desirable properties that are lost on Kolmogorov quotient?
(P.S.: I admit that I've been thinking about pointless topology. There's no tag for it?)