Topologywhere

Topologywhere is a concept in topology that describes a topological space equipped with a parameterized location relation. It provides a framework for modeling how open sets in a space interact with external location-based predicates or constraints, enabling a study of the space under varying conditions.

Formally, a topologywhere on a set X consists of a topology τ on X together with a relation R ⊆ X × P, where P is a parameter space. For each p ∈ P, the fiber Rp = { x ∈ X : (x, p) ∈ R } is required to be an open subset of (X, τ). The collection { Rp : p ∈ P } is called the family of location fibers. In some treatments, one additionally requires that the map p ↦ Rp be continuous when P is endowed with a topology and the Rp are open in τ, yielding a parameter-closedness or upper-semi-continuity flavor.

Examples can be constructed by letting X = R with the standard topology and choosing P = R with the usual topology, and defining Rp = { x ∈ R : x < p }. Then Rp is open in R, and the association p ↦ Rp describes how the open sets vary with the parameter p. Another example uses Rp = (p, ∞) or Rp = [a(p), b(p)) in different parameterizations, illustrating how the same space can carry a family of open sets indexed by P.

Topologywhere relates to parameterized or indexed topologies and shares ideas with fibered constructions, sheaves, and parameterized data analysis. It serves as a lens for examining how spatial structure responds to external constraints, with potential applications in geometry, data analysis, and spatial modeling. See also topology, open set, parameterized topology, and fiber concepts.