Partitionable

Partitionable is a mathematical adjective used to describe objects that can be divided into parts that satisfy a prescribed property. Broadly, the concept relies on partitions: a partition of a set X is a collection of nonempty, pairwise disjoint subsets whose union is X. An object is partitionable with respect to a property P if there exists a partition of its underlying set into parts, each of which has property P.

In practice, the exact meaning of partitionable depends on the field and the property P being considered. For sets, any set is partitionable into singletons, which is a trivial partition. In graph theory, a key example is bipartite graphs: a graph is bipartite precisely when its vertex set can be partitioned into two independent sets. In measure theory or probability, a measurable space may be described as partitionable into measurable atoms or blocks that partition the space and, for instance, carry prescribed measures or probabilities. In topology and geometry, partitionability can refer to decompositions into pieces that satisfy local properties or to structured covers related to partitions of unity, though usage varies by context.

Notes and caveats: partitionability is not a single, universal technical term with one fixed definition. Instead, it denotes the existence of a partition meeting the specified condition, and authors may emphasize different kinds of partitions (finite or infinite, coarse or refined) depending on the problem.

See also: partition of a set, bipartite graphs, partition of unity, measurable partition.