Specialization (pre)order

Specialization (pre)order

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.