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 from X to the power set of Y, written as φ: X→2Y. 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→2Y is said to have a closed graph if the set {(x,y)| y ∈ φ(x)} is a closed subset of X×Y in the product topology.
- Fixed point
- Let φ: X→2X be a set-valued function. Then a ∈ X is a fixed point of φ if a ∈ φ(a).
Read more about this topic: Kakutani Fixed-point Theorem
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