Specialization (pre)order: Mathematics, Topological Space, T1 Separation Axiom, Partially Ordered Set, Denotational Semantics, Order Theory, Domain Algebraic Geometry, Generic Point

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little interest. The specialization order is often considered in applications in computer science, where T0 spaces occur in denotational semantics. The specialization order is also important for identifying suitable topologies on partially ordered sets, as it is done in order theory.