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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
Maps of certain kinds have been given specific names. These include homomorphisms in algebra, isometries in geometry, operators in analysis and representations in group theory. [2] In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. A partial map is a partial function.
Generally, a mapping where the domain and codomain are the same set (or mathematical structure) is a projection if the mapping is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a right inverse. Both notions are strongly related, as follows. Let p be an ...
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
These are examples of cardinal functions, a topic in set-theoretic topology. In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that can be solved using set-theoretic methods, for example, Suslin's problem.
A map with twelve pentagonal faces. In topology and graph theory, a map is a subdivision of a surface such as the Euclidean plane into interior-disjoint regions, formed by embedding a graph onto the surface and forming connected components (faces) of the complement of the graph.
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an ...
If is a topology on and : is any map then set of all that are saturated subsets of forms a topology on . If Y {\displaystyle Y} is also a topological space then f : ( X , τ ) → Y {\displaystyle f:(X,\tau )\to Y} is continuous (respectively, a quotient map ) if and only if the same is true of f : ( X , τ f ) → Y . {\displaystyle f:\left(X ...