Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.
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👉 Continuous function (topology) in the context of Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.
A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.
In this Dossier
Continuous function (topology) in the context of Homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós 'same, similar' and τόπος tópos 'place') if one can be "continuously deformed" into the other, such a deformation being called a homotopy (/həˈmɒtəpiː/ hə-MOT-ə-pee; /ˈhoʊmoʊˌtoʊpiː/ HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
Continuous function (topology) in the context of Kakutani fixed-point theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.
The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics.