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==Definitions==
;Set-valued function: A '''set-valued function''' ''φ'' from the set ''X'' to the set ''Y'' is some rule that associates one ''or more'' points in ''Y'' with each point in ''X''. Formally it can be seen just as an ordinary [[function (mathematics)|function]] from ''X'' to the [[power set]] of ''Y'', written as ''φ'': ''X'' → 2<sup>''Y''</sup>, such that ''φ''(''x'') is non-empty for every <math>x \in X</math>. Some prefer the term '''correspondence''', which is used to refer to a function that for each input may return many outputs. Thus, each element of the ___domain corresponds to a subset of one or more elements of the range.
;Closed graph: A set-valued function φ: ''X'' → 2<sup>''Y''</sup> is said to have a '''closed graph''' if the set {(''x'',''y'') | ''y'' ∈ ''φ''(''x'')} is a [[closed set|closed]] subset of ''X'' × ''Y'' in the [[product topology]] i.e. for all sequences <math>\{x_n\}_{n\in \mathbb{N}}</math> and <math>\{y_n\}_{n\in \mathbb{N}}</math> such that <math>x_n\to x</math>, <math>y_{n}\to y</math> and <math>y_{n}\in
;Fixed point: Let φ: ''X'' → 2<sup>''X''</sup> be a set-valued function. Then ''a'' ∈ ''X'' is a '''fixed point''' of ''φ'' if ''a'' ∈ ''φ''(''a'').
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