neartopological

Neartopological is a term used to describe mathematical frameworks in which nearness between subsets is taken as the fundamental primitive rather than open sets. The idea is to build a structure called a neartopological space from a set X equipped with a nearness relation N that specifies when two subsets are considered close to each other. This approach is viewed by some as a generalization of classical topology, since standard notions such as convergence, continuity, and neighborhood can be recovered from appropriate properties of N.

A neartopological space (X, N) uses the relation of nearness to compare subsets. The nearness relation is required to be compatible with the subset relation and to behave in a way that captures the intuitive idea of closeness; the specifics can vary by author, but the general goal is to enable the study of continuity and convergence without relying on open sets alone. A map f: X → Y between neartopological spaces is called nearness-preserving (continuous in this sense) if A is near B in X implies f(A) is near f(B) in Y. Some treatments also define additional notions such as closure or neighborhoods in terms of the nearness relation.

Relation to classical topology is often illustrated by deriving a neartopological structure from a topological space: for example, defining A near B to mean that the closures of A and B intersect. In this way, the familiar topology can be recovered, while the nearness perspective provides an alternative language for discussing proximity, convergence, and continuity.

Applications and context for neartopology appear in qualitative spatial reasoning, image analysis, sensor networks, and theoretical computer science, particularly in settings where a metric is unavailable or undesirable. The term is used in niche mathematical literature and should be understood as part of a family of related concepts, including proximity spaces and nearness spaces, which formalize similar ideas through different axiomatizations.