Topology Notes: Ordered Spaces
نویسنده
چکیده
Consider the following properties of a binary relation ≤ on a set X: (Reflexivity) For all x ∈ X, x ≤ x. (Anti-Symmetry) For all x, y ∈ X, if x ≤ y and y ≤ x, then x = y. (Transitivity) For all x, y, z ∈ X, if x ≤ y and y ≤ z, then x ≤ z. (Totality) For all x, y ∈ X, either x ≤ y or y ≤ x. A relation which satisfies reflexivity and transitivity is called a quasi-ordering. A relation which satisfies reflexivity, anti-symmetry and transitivity is called a partial ordering. A relation which satisfies all four properties is called an ordering (sometimes a total or linear ordering). An ordered set is a pair (X,≤) where X is a set and ≤ is an ordering on X.
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