Editing Kolmogorov space (section)

==Operating with T<sub>0</sub> spaces ==

Commonly studied topological spaces are all T<sub>0</sub>.
Indeed, when mathematicians in many fields, notably [[analysis (mathematics)|analysis]], naturally run across non-T<sub>0</sub> spaces, they usually replace them with T<sub>0</sub> spaces, in a manner to be described below. To motivate the ideas involved, consider a well-known example. The space [[Lp space|L<sup>2</sup>('''R''')]] is meant to be the space of all [[measurable function]]s ''f'' from the [[real line]] '''R''' to the [[complex plane]] '''C''' such that the [[Lebesgue integral]] of |''f''(''x'')|<sup>2</sup> over the entire real line is [[finite set|finite]].
This space should become a [[normed vector space]] by defining the norm ||''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 [[0 (number)|zero]].
The standard solution is to define L<sup>2</sup>('''R''') to be a set of [[equivalence class]]es of functions instead of a set of functions directly.
This constructs a [[Quotient space (topology)|quotient space]] of the original seminormed vector space, and this quotient is a normed vector space. It inherits several convenient properties from the seminormed space; see below.

In general, when dealing with a fixed topology '''T''' on a set ''X'', it is helpful if that topology is T<sub>0</sub>. On the other hand, when ''X'' is fixed but '''T''' is allowed to vary within certain boundaries, to force '''T''' to be T<sub>0</sub> may be inconvenient, since non-T<sub>0</sub> topologies are often important special cases. Thus, it can be important to understand both T<sub>0</sub> and non-T<sub>0</sub> versions of the various conditions that can be placed on a topological space.