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The T0 condition is one of the separation axioms.
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 one of the following equivalent conditions holds:
Otherwise, x and y are said to be topologically distinguishable. Loosely speaking, this means that the topology on X is capable of distinguishing between x and y.
In an indiscrete space, for example, any two points are topologically indistinguishable.
Note that the closure of a point x contains all points indistinguishable from x. It may contain other points as well.
The definition of a T0 space is now simple; X is T0 if and only if every pair of distinct points is topologically distinguishable.
Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton sets {x} and {y} are separated, then the points x and y must be topologically distinguishable. That is,
The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated. In a T0 space, the second arrow above reverses; points are distinct iff they are distinguishable. This is how the T0 axiom fits in with the rest of the separation axioms.
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. This characterisation should be contrasted with an analogous characterisation of T1 spaces, where one can specify beforehand which points will belong to the open set.
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)In mathematics, the Lp and spaces are spaces of p-power integrable functions and corresponding sequence spaces''. They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. See also root mean square is meant to be the space of all measurable functionIn mathematics, measurable functions are well-behaved functions between measurable spaces. Functions studied in analysis that are not measurable are generally considered pathological. If X is a σ-algebra over S and Y is a σ-algebra over T thens f from the real lineIn mathematics, the real line is simply the set of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space. The real line has been studied at least since the days of t R to the complex plane C such that the Lebesgue integral of |f(x)|2 over the entire real line not only exists but also 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 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.