Editing Kolmogorov space (section)

== The Kolmogorov quotient ==

Topological indistinguishability of points is an [[equivalence relation]]. No matter what topological space ''X'' might be to begin with, the [[Quotient space (topology)|quotient space]] under this equivalence relation is always T<sub>0</sub>. This quotient space is called the '''Kolmogorov quotient''' of ''X'', which we will denote KQ(''X''). Of course, if ''X'' was T<sub>0</sub> to begin with, then KQ(''X'') and ''X'' are [[natural (category theory)|natural]]ly [[homeomorphic]].
Categorically, Kolmogorov spaces are a [[reflective subcategory]] of topological spaces, and the Kolmogorov quotient is the reflector.

Topological spaces ''X'' and ''Y'' are '''Kolmogorov equivalent''' when their Kolmogorov quotients are homeomorphic. Many properties of topological spaces are preserved by this equivalence; that is, if ''X'' and ''Y'' are Kolmogorov equivalent, then ''X'' has such a property if and only if ''Y'' does.
On the other hand, most of the ''other'' properties of topological spaces ''imply'' T<sub>0</sub>-ness; that is, if ''X'' has such a property, then ''X'' must be T<sub>0</sub>.
Only a few properties, such as being an [[indiscrete space]], are exceptions to this rule of thumb.
Even better, many [[structure (mathematics)|structure]]s defined on topological spaces can be transferred between ''X'' and KQ(''X'').
The result is that, if you have a non-T<sub>0</sub> topological space with a certain structure or property, then you can usually form a T<sub>0</sub> space with the same structures and properties by taking the Kolmogorov quotient.

The example of L<sup>2</sup>('''R''') displays these features.
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 space|pseudometric]] and a [[uniform structure]] that are compatible with the topology.
Also, there are several properties of these structures; for example, the seminorm satisfies the [[parallelogram identity]] and the uniform structure is [[complete space|complete]].  The space is not T<sub>0</sub> since any two functions in L<sup>2</sup>('''R''') that are equal [[almost everywhere]] are indistinguishable with this topology.
When we form the Kolmogorov quotient, the actual L<sup>2</sup>('''R'''), these structures and properties are preserved.
Thus, L<sup>2</sup>('''R''') is also a complete seminormed vector space satisfying the parallelogram identity.
But we actually get a bit more, since the space is now T<sub>0</sub>.
A seminorm is a norm if and only if the underlying topology is T<sub>0</sub>, so L<sup>2</sup>('''R''') is actually a complete normed vector space satisfying the parallelogram identity&mdash;otherwise known as a [[Hilbert space]].
And it is a Hilbert space that mathematicians (and [[physicists]], in [[quantum mechanics]]) generally want to study.  Note that the notation L<sup>2</sup>('''R''') usually denotes the Kolmogorov quotient, the set of [[equivalence class]]es 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.