mapset

Mapset is a term used primarily in mathematics to denote a set whose elements are maps (functions) between fixed sets. If X and Y are sets, Map(X,Y) denotes the collection of all functions from X to Y; a mapset S is any subset of Map(X,Y) or a collection of such functions that satisfy a chosen property (for example, continuity or linearity). The notation Y^X is also used for Map(X,Y).

Examples and variants: If X has n elements and Y has m elements, there are m^n distinct maps X→Y, so |Map(X,Y)| = m^n. If X is a topological space and Y a topological space, the subset of continuous maps C(X,Y) is a common mapset. In topology and analysis, mapping spaces can be equipped with topologies such as the compact-open topology, making Map(X,Y) a topological space.

Relation to computer science: In programming contexts, map usually means a data structure that associates keys with values, while a set stores distinct elements. A "mapset" can refer to a collection of maps used to describe configurations, state spaces, or search spaces, rather than a standard data type. Typical operations include adding or removing maps, testing membership, and applying a map to an input when domains align.

In category theory and type theory, maps are morphisms; the collection of morphisms from X to Y is often denoted Hom(X,Y). In some contexts this is called Map(X,Y). The concept of a mapset helps formalize families of functions and their properties, and it is a basic construct in combinatorics, model theory, and logic.

See also: function, function space, mapping, hom-set, mapping space, data structure.