Closed graph

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.^[1]^[2] A function $f : X \to Y$ between topological spaces has a closed graph if its graph is a closed subset of the product space $X \times Y$ . A related property is open graph.^[3]

This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.

Definitions[]

Graphs and multifunctions[]

Definition and notation: The graph of a function

f : X \to Y

is the set

Gr f := { (x, f (x)) : x \in X} = { (x, y) \in X \times Y : y = f (x) }

.

Notation: If

Y

is a set then the power set of

Y

, which is the set of all subsets of

Y

, is denoted by

2 Y

or

[1]

[2]

[3]