In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain 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 t ...

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

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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

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

References

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