Definition and Motivation
Consider any topological space X. The specialization preorder ≤ on X is defined by setting
- x ≤ y if and only if cl{x} is a subset of cl{y},
where cl{x} denotes the closure of the singleton set {x}, i.e. the intersection of all closed sets containing {x}. While this brief definition is convenient, it is helpful to note that the following statement is equivalent:
- x ≤ y if and only if y is contained in all open sets that contain x.
This definition explains why one speaks of a "specialization": y is more special than x, since it is contained in more open sets. This is particularly intuitive if one views open sets as properties that a point x may or may not have. The more open sets contain a point, the more properties it has, and the more special it is. The usage is consistent with the classical logical notions of genus and species; and also with the traditional use of generic points in algebraic geometry. Specialization as an idea is applied also in valuation theory.
The intuition of upper elements being more specific is typically found in domain theory, a branch of order theory that has ample applications in computer science.
Read more about this topic: Specialization (pre)order
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