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Topological indistinguishability
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 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. (See Hausdorff's axiomatic neighborhood systems.)
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.
Topological indistinguishability defines an equivalence relation on any topological space X. If x and y are points of X we write x ≡ y for "x and y are topologically indistinguishable". The equivalence class of x will be denoted by [x].
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. In fact, almost all of the examples given below are completely regular.
The topological indistinguishability relation on a space X can be recovered from a natural preorder on X called the specialization preorder. For points x and y in X this preorder is defined by
where cl{y} denotes the closure of {y}. Equivalently, x ≤ y if the neighborhood system of x, denoted Nx, is contained in the neighborhood system of y:
It is easy to see that this relation on X is reflexive and transitive and so defines a preorder. In general, however, this preorder will not be antisymmetric. Indeed, the equivalence relation determined by ≤ is precisely that of topological indistinguishability:
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Topological indistinguishability
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 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. (See Hausdorff's axiomatic neighborhood systems.)
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.
Topological indistinguishability defines an equivalence relation on any topological space X. If x and y are points of X we write x ≡ y for "x and y are topologically indistinguishable". The equivalence class of x will be denoted by [x].
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. In fact, almost all of the examples given below are completely regular.
The topological indistinguishability relation on a space X can be recovered from a natural preorder on X called the specialization preorder. For points x and y in X this preorder is defined by
where cl{y} denotes the closure of {y}. Equivalently, x ≤ y if the neighborhood system of x, denoted Nx, is contained in the neighborhood system of y:
It is easy to see that this relation on X is reflexive and transitive and so defines a preorder. In general, however, this preorder will not be antisymmetric. Indeed, the equivalence relation determined by ≤ is precisely that of topological indistinguishability: