Closed graph property

"Open graph" redirects here. For the Facebook API, see Open Graph Protocol.

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A related property is open graph.

Definition

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 $\{x_{n}\}_{n\in \mathbb {N} }$ and $\{y_{n}\}_{n\in \mathbb {N} }$ such that $x_{n}\to x$, $y_{n}\to y$ and $y_{n}\in \varphi (x_{n})$ for all $n$, we have $y\in \varphi (x)$.

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.

Characterization

The closed graph theorem for set-valued functions 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.

See also

• Closed graph theorem
• Kakutani fixed-point theorem