Closed graph

A set-valued function φ: X → 2^Y is said to have a closed graph if the set {(x,y) | y ∈ φ(x)} is a closed subset of X × Y in its product topology. In other words, for all sequences ${\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}$ and ${\displaystyle \{y_{n}\}_{n\in \mathbb {N} }}$ such that ${\displaystyle x_{n}\to x}$, ${\displaystyle y_{n}\to y}$ and ${\displaystyle y_{n}\in \varphi (x_{n})}$ for all ${\displaystyle n}$, we have ${\displaystyle y\in \varphi (x)}$.

Similarly, φ: X → 2^Y is said to have an open graph if the set {(x,y) | y ∈ φ(x)} is an open subset of X × Y in its product topology.

The closed graph theorem for set-valued functions[4] says that, for a compact Hausdorff range space Y, a set-valued function φ : X → 2^Y has a closed graph if and only if it is upper hemicontinuous and φ(x) is a closed set for all x.

• Closed graph theorem
• Kakutani fixed-point theorem