Such spaces naturally occur in computer science, specifically in denotational semantics.
On the other hand, if the singleton sets {x} and {y} are separated then the points x and y must be topologically distinguishable.
That is, The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated.
The space L2(R) is meant to be the space of all measurable functions f from the real line R to the complex plane C such that the Lebesgue integral of |f(x)|2 over the entire real line is finite.
On the other hand, when X is fixed but T is allowed to vary within certain boundaries, to force T to be T0 may be inconvenient, since non-T0 topologies are often important special cases.
From the point of view of topology, the seminormed vector space that we started with has a lot of extra structure; for example, it is a vector space, and it has a seminorm, and these define a pseudometric and a uniform structure that are compatible with the topology.
When we form the Kolmogorov quotient, the actual L2(R), these structures and properties are preserved.
Thus, L2(R) is also a complete seminormed vector space satisfying the parallelogram identity.
And it is a Hilbert space that mathematicians (and physicists, in quantum mechanics) generally want to study.
Note that the notation L2(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrable functions that differ on sets of measure zero, rather than simply the vector space of square integrable functions that the notation suggests.
In general, it is possible to define non-T0 versions of both properties and structures of topological spaces.
This is a sensible, albeit less famous, property; in this case, such a space X is called preregular.
The T0 requirement can be added or removed arbitrarily using the concept of Kolmogorov quotient.