(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?)
Here is an example of a space of genuine interest that is not $\text{T}_0$, and for which the non-$\text{T}_0$ aspects are rich and significant. The answer is long, in an attempt to be readable.
Let $\mathcal{C}$ be the collection of all properly convex (i.e. convex and closure contained in an affine chart) open subsets of $\mathbb{RP}^2$. Give $\mathcal{C}$ the Hausdorff topology on closures with respect to a reference metric on $\mathbb{RP}^2$. That is, $\Omega_1, \Omega_2$ are close if every point in $\overline{\Omega_1}$ is close to a point of $\overline{\Omega_2}$ and vice versa. The group $\text{SL}_3(\mathbb{R})$ of matrices acts on $\mathbb{RP}^2$ and sends lines to lines because linear transformations send subspaces to subspaces. It follows that $\text{SL}_3(\mathbb{R})$ preserves proper convexity, and acts continuously on $\mathcal{C}$. Let $\mathfrak{C}$ be $\mathcal{C}/\text{SL}_3(\mathbb{R})$, equipped with the quotient topology.
$\mathfrak{C}$ is not $\text{T}_0$. The standard proof $\mathfrak{C}$ is not $\text{T}_1$ by examining matricies of the form $A_t = \text{diag}(e^t, 1, e^{-t})$. Any such matrix preserves an ellipse $E$, and for large $t$ takes small neighborhoods of a boundary point $p^-$ onto all of $E$ except a small neighborhood of a second boundary point $p^+$. Taking a half-ellipse $C$ containing $p^-$, we then have $[A_t C] = [C]$ for all $t$, but $A_t C$ converges to $E$ in the Hausdorff topology, so that the constant sequence $[C]$ has distinct limit points $[C]$ and $[E]$. Non-$\text{T}_0$-ness is less easy to see.
The non-$\text{T}_0$ structure of $\mathfrak{C}$ is both rich and controlled: there are dense points in $\mathfrak{C}$, while there finite-dimensional families of arbitrarily large dimension of closed points in $\mathfrak{C}$. On the other hand, if one works with pointed properly convex domains, the analogous space $\mathfrak{C}$ is compact and Hausdorff (J. P. Benzecri's thesis and Goldman's 1990 paper "Convex real projective structures..." are standard sources for this).
So why care about this non-$\text{T}_0$ space rather than its Kolmogorov quotient? Well, it's much more common in practice to care about convex domains than mysterious equivalence classes of convex sets. Sometimes math just gives you lemons and the natural topology on the space you care about isn't $\text{T}_0$.
More significantly, the non-$\text{T}_0$-ness of $\mathfrak{C}$ is a reflection of orbit structure of dynamics of an interesting action of $\text{SL}_3(\mathbb{R})$. It's also the case that geometrically interesting domains have special behavior in $\mathfrak{C}$, which influences their theory.
Let me give an example. Domains with so much symmetry to admit a cocompact action by $\text{SL}_3(\mathbb{R})$ ("divisible domains") are closed points of $\mathfrak{C}$ and are of intense interest. There are many such domains with $C^1$ boundary, with the only such domain with $C^2$ boundary the ellipse. Any divisible domain in $\mathbb{RP}^2$ with $C^1$ but not $C^2$ boundary contains a dense set of $C^2$ boundary points, all of which have curvature $0$. The standard proof that any $C^2$ boundary point of a non-ellipse boundary domain has zero curvature uses that divisible domains are closed points in $\mathfrak{C}$. The sketch is that if $\Omega$ had a $C^2$ boundary point, then using an osculating ellipse and a family of diagonal matricies, one constructs the ellipse in the $\mathfrak{C}$-closure of $\{[\Omega]\}$, which is impossible. One may similarly prove the fact I mentioned above that the ellipse is the only $C^2$ divisible domain in $\mathbb{RP}^2$.
This is of course only "actually" using the non-$\text{T}_1$ structure of $\mathfrak{C}$. For objects that witness meaningful non-$\text{T}_0$ structure of $\mathfrak{C}$, certain beautiful (Hitchin) representations of surface groups in $\text{SL}_4(\mathbb{R})$ turn up interesting phenomena. Such representations $\rho$ preserve domains $\Omega \subset \mathbb{RP}^3$ foliated by peculiar convex domains $\{C_t\}$ $(t \in S^1)$ in copies of $\mathbb{RP}^2$. It turns out for all except one small class of $\rho$, the families $\{[C_t]\}_{t \in S^1}$ give the only known examples of closed subsets of $\mathfrak{C}$ that are minimal among nonempty closed subsets of $\mathfrak{C}$ but not points, which is a truly-$\text{T}_0$ phenomenon (the source is "Leaves of Foliated..." by me). One may then deduce similar things about these domains as the above with similar arguments to the divisible case.