Cocentraltopologies

Cocentraltopologies is a term used in some corners of topology to denote a class of topological structures defined by a distinguished family of subsets called cocenters. The idea is to emphasize a dual or dualized-constructive role for a chosen family of sets in generating the topology, by viewing these cocenters as the primary building blocks from which open sets arise.

Definition. Let X be a set. A cocentraltopology on X is a topology T for which there exists a family C of subsets of X, the cocenters, such that:

- C forms a basis for T: every open set is a union of elements of C, and for any x in A ∩ B with A,B ∈ C, there is C0 ∈ C with x ∈ C0 ⊆ A ∩ B.

- X ∈ C (often included to ensure X itself is recognizable as a cocenter).

In this framework, T is precisely the topology generated by the cocenter family C. Different authors may impose additional structure on C, such as finite generation, closure under finite intersections, or that C arises from a partition of X.

Basic properties. Because T is generated by C as a basis, many standard facts about bases apply. If C is finite or if C has extra closure properties, one can obtain specialized separation axioms, compactness, or connectedness criteria expressed in terms of C. The notion is intentionally flexible: without extra constraints, every topology on X can be described by some suitable C, while extra conditions on C yield interesting subfamilies.

Examples. If C consists of all singletons on a finite X, T is the discrete topology. If C = {X}, T is the indiscrete topology. If C is the family of blocks of a partition of X, T is the corresponding partition topology.

Relation to other notions. Cocentraltopologies aligns with the standard idea of defining a topology by a basis, but it foregrounds a chosen cocenter family as a central organizing tool. It is used mainly in exploratory or expository contexts and is subject to variation in definitions across sources.