In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a coincidence point (or simply coincidence) of two mappings

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 is a point in their domain

Domain (mathematics)

In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

 having the same image point under both mappings.

Formally, given two mappings
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

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

 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 mapping.

Just as fixed point theory has its fixed-point theorem

Fixed-point theorem

In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point , under some conditions on F that can be stated in general terms...

s, there are theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

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

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s, is the Lefschetz coincidence theorem

Lefschetz fixed-point theorem

In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...

, which is typically known only in its special case formulation for fixed points.

Coincidence points, like fixed points, are today studied using many tools from mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 and topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

. An Equaliser is a generalization of the coincidence set.