Definition
A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. That is, for any two different points x and y there is an open set which contains one of these points and not the other.
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,
- separated ⇒ topologically distinguishable ⇒ distinct
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 if and only if they are distinguishable. This is how the T0 axiom fits in with the rest of the separation axioms.
Read more about this topic: Kolmogorov Space
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