Kolmogorov Space - The Kolmogorov Quotient

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 under this equivalence relation is always T₀. This quotient space is called the Kolmogorov quotient of X, which we will denote KQ(X). Of course, if X was T₀ to begin with, then KQ(X) and X are naturally 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 T₀-ness; that is, if X has such a property, then X must be T₀. Only a few properties, such as being an indiscrete space, are exceptions to this rule of thumb. Even better, many structures defined on topological spaces can be transferred between X and KQ(X). The result is that, if you have a non-T₀ topological space with a certain structure or property, then you can usually form a T₀ space with the same structures and properties by taking the Kolmogorov quotient.

The example of L2(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 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. The space is not T₀ since any two functions in L2(R) which are equal almost everywhere are indistinguishable with this 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. But we actually get a bit more, since the space is now T₀. A seminorm is a norm if and only if the underlying topology is T₀, so L2(R) is actually a complete normed vector space satisfying the parallelogram identity — 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 L2(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrable functions which differ on sets of measure zero, rather than simply the vector space of square integrable functions which the notation suggests.