Kolmogorov space

To define T₀ spaces, we first define the concept of topologically distinguishable points. If X is a topological space and x and y are points in X, then x and y are topologically indistinguishable if and only if they have exactly the same neighbourhoods. Otherwise, they are topologically distinguishable. The definition of a T₀ space is now simple; X is T₀ if and only if every pair of distinct points is topologically distinguishable.

This definition may also be formulated as follows: X is a T₀ space if and only if for any two distinct points in X there exists an open subset of X which contains one of the points but not the other.

Almost every topological space studied in ordinary mathematics is T₀. Indeed, when mathematicians in many fields, notably analysis, naturally run across non-T₀ spaces, they usually replace them with T₀ spaces, in a manner described below.

In general, when dealing with a fixed topology T on a set X, it's helpful if that topology is T₀. On the other hand, when X is fixed but T is allowed to vary within certain boundaries, it can be annoying to force T to be T₀, since the non-T₀ topologies are often important special cases. Thus, it can be important to understand both T₀ and non-T₀ versions of the various conditions that can be placed on a topological space.

To motivate the ideas involved, let's consider a well known example. 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 ∫_-∞^∞ |f(x)|² dx exists and is finite. This space should become a normed vector space by defining ||f|| to be the square root of that integral. The problem is that this is not really a norm, only a seminorm[?], because there are functions other than the zero function[?] whose (semi)norms are zero. The standard solution is to define L²(R) to be a set of equivalence classes of functions instead of a set of functions directly. This constructs a quotient space of the original seminormed vector space, and this quotient is a normed vector space. It inherits several nice properties from the seminormed space, as we will see below.

Both the problem and the solution are reflected at the level of the topologies defined by the norm and seminorm. If a function's seminorm is zero, then it's topologically indistinguishable from the zero function. More generally, functions are identified in the construction of the quotient space precisely when they are topologically indistinguishable in the original seminormed space.

The general construction is 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 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.

Topological spaces X and Y are Kolmogorov equivalent iff their Kolmogorov quotients are homeomorphic. The nice thing about Kolmogorov equivalence is that 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 iff 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 nice 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's 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. 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 iff 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) want to study.

You may notice that, although norms were defined first, people came up with the definition of seminorm as well, which is a sort of non-T₀ version of a norm. In general, it is possible to define non-T₀ versions of both properties and structures of topological spaces. First, consider a property of toplogical spaces, such as being Hausdorff. One can then define another property of topological spaces by defining the space X to satisfy the property if and only if the Kolmogorov quotient KQ(X) is Hausdorff. This is a sensible, albeit less famous, property; in this case, such a space X is called preregular. (There even turns out to be a more direct definition of preregularity.) Now consider a structure that can be placed on topological spaces, such as a metric. We can define a new structure on topological spaces by letting an example of the structure on X be simply a metric on KQ(X). This is a sensible structure on X; it is a pseudometric. (Again, there is a more direct definition of pseudometric.)

In this way, there is a natural way to remove T₀-ness from the requirements for a property or structure. It's generally easier to study spaces that are T₀, but it may also be easier to allow structures that aren't T₀ to get a fuller picture. The T₀ requirement can be added or removed at will, using the concept of Kolmogorov quotient.