resolvability

Resolvability is a mathematical concept used to describe the ability to distinguish or separate elements within a structure according to a prescribed criterion. The term appears in several areas of mathematics, most notably topology and graph theory, where it is formalized in different ways that share the underlying idea of partitioning or distinguishing components.

In topology, a space is called resolvable if it can be partitioned into two disjoint dense subsets. More generally, a space is κ-resolvable if it can be partitioned into κ dense, pairwise disjoint subsets. Many standard spaces are at least two-resolvable, including the real line with its usual topology, which can be split into the dense sets of rational and irrational numbers. The notions of maximal resolvability and extraresolvability extend these ideas to larger cardinals under various set-theoretic assumptions.

In graph theory, resolvability refers to the use of resolving sets. A set S of vertices in a connected graph G is resolving if every vertex of G has a unique distance vector to S, enabling its identification by measurements to S. The metric dimension of G is the smallest size of a resolving set. Examples include path graphs P_n, whose metric dimension is 1, and cycles C_n with metric dimension 2. The complete graph K_n has metric dimension n–1. Computing the metric dimension is generally NP-hard, and resolving sets have applications in network navigation, chemistry, and combinatorial optimization.

Resolvability thus captures a unifying idea across disciplines: the possibility of distinguishing elements through a structured partition or by distance-based profiles. Variants and refinements continue to be studied in topology and graph theory, reflecting ongoing interest in how local measurements determine global structure.