Kolmogorov Space - Operating With T₀ Spaces

Operating With T₀ Spaces

Examples of topological space typically studied are 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 to be described below. To motivate the ideas involved, 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 of |f(x)|2 over the entire real line is 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 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 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 T₀. On the other hand, when X is fixed but T is allowed to vary within certain boundaries, to force T to be T₀ may be inconvenient, since 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.