Editing Kolmogorov space (section)

== Removing T<sub>0</sub> ==

Although norms were historically defined first, people came up with the definition of seminorm as well, which is a sort of non-T<sub>0</sub> version of a norm. In general, it is possible to define non-T<sub>0</sub> versions of both properties and structures of topological spaces. First, consider a property of topological spaces, such as being [[Hausdorff space|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 space|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 space|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 space|pseudometric]]. (Again, there is a more direct definition of pseudometric.)

In this way, there is a natural way to remove T<sub>0</sub>-ness from the requirements for a property or structure. It is generally easier to study spaces that are T<sub>0</sub>, but it may also be easier to allow structures that aren't T<sub>0</sub> to get a fuller picture. The T<sub>0</sub> requirement can be added or removed arbitrarily using the concept of Kolmogorov quotient.