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A map is a function, as in the association of any of the four colored shapes in X to its color in Y. In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2]
Projection (mathematics) – Mapping equal to its square under mapping composition; Projection (measure theory) Projection (linear algebra) – Idempotent linear transformation from a vector space to itself; Projection (relational algebra) – Operation that restricts a relation to a specified set of attributes
Every set is a projective object in Set (assuming the axiom of choice). The finitely presentable objects in Set are the finite sets. Since every set is a direct limit of its finite subsets, the category Set is a locally finitely presentable category. If C is an arbitrary category, the contravariant functors from C to Set are often an important ...
Other authors call a map proper if it is continuous and closed with compact fibers; that is if it is a continuous closed map and the preimage of every point in is compact. The two definitions are equivalent if Y {\displaystyle Y} is locally compact and Hausdorff .
In the sense of universal algebra, a pointed set is a set together with a single nullary operation:, [a] which picks out the basepoint. [7] Pointed maps are the homomorphisms of these algebraic structures. The class of all pointed sets together with the class of all based maps forms a category.
Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with " = {\displaystyle =} " and " ∈ {\displaystyle \in } " of classical set theory is usually used, so this is not to be confused with a constructive types approach.
Cardinal functions are widely used in topology as a tool for describing various topological properties. [4] [5] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [6] prefer to define the cardinal functions listed below so that they never take on finite cardinal numbers as values; this requires modifying some of the definitions ...
In mathematics, if is a subset of , then the inclusion map is the function that sends each element of to , treated as an element of ::, =. An inclusion map may also be referred to as an inclusion function , an insertion , [ 1 ] or a canonical injection .