Topological Indistinguishability

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are points in X, and A is the set of all neighborhoods which contain x, and B is the set of all neighborhoods which contain y, then x and y are "topologically indistinguishable" if and only if A=B. Intuitively, two points are topologically indistinguishable if the topology of X is unable to discern between the points. Two points of X are topologically distinguishable if they are not topologically indistinguishable. This means there is an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely one of the two points). This open set can then be used to distinguish between the two points. A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. This is the weakest of the separation axioms.