mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a coincidence point (or simply coincidence) of two functions is a point in their common

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

having the same image. Formally, given two functions :

f,g \colon X \rightarrow Y

we say that a point ''x'' in ''X'' is a ''coincidence point'' of ''f'' and ''g'' if ''f''(''x'') = ''g''(''x''). Coincidence theory (the study of coincidence points) is, in most settings, a generalization of fixed point theory, the study of points ''x'' with ''f''(''x'') = ''x''. Fixed point theory is the special case obtained from the above by letting ''X = Y'' and taking ''g'' to be the

identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...

. Just as fixed point theory has its

fixed-point theorem In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors cla ...

s, there are

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

s that guarantee the existence of coincidence points for pairs of functions. Notable among them, in the setting of

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s, is the Lefschetz coincidence theorem, which is typically known only in its

special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is ...

formulation for fixed points. Coincidence points, like fixed points, are today studied using many tools from

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

and

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

. An equaliser is a generalization of the coincidence set..

References

Mathematical analysis Topology Fixed points (mathematics) {{analysis-stub