Kolmogorov Space - Removing T₀

Removing T₀

Although norms were historically 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 topological 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 is 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 arbitrarily using the concept of Kolmogorov quotient.