That is, if x and y are points in X, and Nx is the set of all neighborhoods that contain x, and Ny is the set of all neighborhoods that contain y, then x and y are "topologically indistinguishable" if and only if Nx = Ny.
By definition, any two distinct points in a T0 space are topologically distinguishable.
On the other hand, regularity and normality do not imply T0, so we can find nontrivial examples of topologically indistinguishable points in regular or normal topological spaces.
Topological indistinguishability is better behaved in these spaces and easier to understand.
There are several equivalent ways of determining when two points are topologically indistinguishable.
Then the following statements are equivalent: These conditions can be simplified in the case where X is symmetric space.
The space KX is called the Kolmogorov quotient or T0 identification of X.
The space KX is, in fact, T0 (i.e. all points are topologically distinguishable).
Intuitively, the Kolmogorov quotient does not alter the topology of a space.