Proximitii

Proximitii is a term used in topology to describe a family of proximity relations that express the notion of nearness between subsets of a common set. Each proximitium δ defines, for any two subsets A and B of a given space X, whether A and B are near. The collection of such relations, when organized as a family, is referred to as proximitii.

Proximitii generalize the classical concept of a proximity relation. They typically satisfy symmetry (A δ B implies B δ A), monotonicity (if A ⊆ A' and B ⊆ B' and A δ B then A' δ B'), and compatibility with closure in the underlying topology. In a standard formulation, a proximity relation δ on a topological space X is linked to the closure operator by defining A δ B to hold when the closures of A and B intersect. Different proximitii may be parameterized by a space, a scale, or a modeling assumption, allowing a single underlying X to carry multiple “nearness” structures simultaneously.

Examples and applications: In Euclidean spaces with the usual topology, one common proximitium is A δ B if closure(A) ∩ closure(B) ≠ ∅. Proximitii have found use in domains where quantitative distances are unavailable or undesirable, such as qualitative spatial reasoning, image analysis, clustering, and geographic information systems, where a notion of nearness drives algorithms and decision-making. They also relate to other generalized nearness concepts such as proximity spaces and nearness relations.

Terminology: proximitii is a plural form used in some texts to distinguish multiple proximity relations on the same underlying set; in more common usage these are simply called proximity relations or proximity structures.