Closed graph property

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 set-valued functions

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

𝒫(Y)

.

Definition: If

X

and

Y

are sets, a set-valued function in

Y

on

X

(also called a

Y

-valued multifunction on

X

) is a function

F : X \to 2 Y

with domain

X

that is valued in

2 Y

. That is,

F

is a function on

X

such that for every

x \in X

,

F (x)

is a subset of

Y

.

Some authors call a function $F : X \to 2 Y$ a set-valued function only if it satisfies the additional requirement that $F (x)$ is not empty for every $x \in X$ ; this article does not require this.

Definition and notation: If

F : X \to 2 Y

is a set-valued function in a set

Y

then the graph of

F

is the set

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

.

Definition: A function

f : X \to Y

can be canonically identified with the set-valued function

F : X \to 2 Y

defined by

F (x) := {f (x) }

for every

x \in X

, where

F

is called the canonical set-valued function induced by (or associated with)

f

.

Note that in this case, $Gr f = Gr F$ .

Open and closed graph

We give the more general definition of when a $Y$ -valued function or set-valued function defined on a subset $S$ of $X$ has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace $S$ of a topological vector space $X$ (and not necessarily defined on all of $X$ ). This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis.

Assumptions: Throughout,

X

and

Y

are topological spaces,

S \subseteq X

, and

f

is a

Y

-valued function or set-valued function on

S

(i.e.

f : S \to Y

or

f : S \to 2 Y

).

X \times Y

will always be endowed with the product topology.

Definition:^[4] We say that

f

has a closed graph (resp. open graph, sequentially closed graph, sequentially open graph) in $X \times Y$ if the graph of

f

,

Gr f

, is a closed (resp. open, sequentially closed, sequentially open) subset of

X \times Y

when

X \times Y

is endowed with the product topology. If

S = X

or if

X

is clear from context then we may omit writing "in

X \times Y

"

Observation: If

g : S \to Y

is a function and

G

is the canonical set-valued function induced by

g

(i.e.

G : S \to 2 Y

is defined by

G (s) := {g (s) }

for every

s \in S

) then since

Gr g = Gr G

,

g

has a closed (resp. sequentially closed, open, sequentially open) graph in

X \times Y

if and only if the same is true of

G

.

Closable maps and closures

Definition: We say that the function (resp. set-valued function)

f

is closable in $X \times Y$ if there exists a subset

D \subseteq X

containing

S

and a function (resp. set-valued function)

F : D \to Y

whose graph is equal to the closure of the set

Gr f

in

X \times Y

. Such an

F

is called a closure of $f$ in $X \times Y$ , is denoted by

f

, and necessarily extends

f

.

Additional assumptions for linear maps: If in addition, $S$ , $X$ , and $Y$ are topological vector spaces and $f : S \to Y$ is a linear map then to call $f$ closable we also require that the set $D$ be a vector subspace of $X$ and the closure of $f$ be a linear map.

Definition: If

f

is closable on

S

then a core or essential domain of

f

is a subset

D \subseteq S

such that the closure in

X \times Y

of the graph of the restriction

f | D : D \to Y

of

f

to

D

is equal to the closure of the graph of

f

in

X \times Y

(i.e. the closure of

Gr f

in

X \times Y

is equal to the closure of

Gr f | D

in

X \times Y

).

Closed maps and closed linear operators

Definition and notation: When we write

f : D (f) \subseteq X \to Y

then we mean that

f

is a

Y

-valued function with domain

D (f)

where

D (f) \subseteq X

. If we say that

f : D (f) \subseteq X \to Y

is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) then we mean that the graph of

f

is closed (resp. sequentially closed) in

X \times Y

(rather than in

D (f) \times Y

).

When reading literature in functional analysis, if $f : X \to Y$ is a linear map between topological vector spaces (TVSs) (e.g. Banach spaces) then " $f$ is closed" will almost always means the following:

Definition: A map

f : X \to Y

is called closed if its graph is closed in

X \times Y

. In particular, the term "closed linear operator" will almost certainly refer to a linear map whose graph is closed.

Otherwise, especially in literature about point-set topology, " $f$ is closed" may instead mean the following:

Definition: A map

f : X \to Y

between topological spaces is called a closed map if the image of a closed subset of

X

is a closed subset of

Y

.

These two definitions of "closed map" are not equivalent. If it is unclear, then it is recommended that a reader check how "closed map" is defined by the literature they are reading.

Characterizations

Throughout, let $X$ and $Y$ be topological spaces.

Function with a closed graph

If $f : X \to Y$ is a function then the following are equivalent:

$f$ has a closed graph (in $X \times Y$ );
(definition) the graph of $f$ , $Gr f$ , is a closed subset of $X \times Y$ ;
for every x ∈ X and net x_• = (x_i)_{i ∈ I} in X such that x_• → x in X, if y ∈ Y is such that the net f(x_•) := (f(x_i))_{i ∈ I} → y in Y then y = f(x);^[4]
- Compare this to the definition of continuity in terms of nets, which recall is the following: for every $x \in X$ and net $x • = (x i) i \in I$ in $X$ such that $x • \to x$ in $X$ , $f (x •) \to f (x)$ in $Y$ .
- Thus to show that the function $f$ has a closed graph we may assume that $f (x •)$ converges in $Y$ to some $y \in Y$ (and then show that $y = f (x)$ ) while to show that $f$ is continuous we may not assume that $f (x •)$ converges in $Y$ to some $y \in Y$ and we must instead prove that this is true (and moreover, we must more specifically prove that $f (x •)$ converges to $f (x)$ in $Y$ ).

and if $Y$ is a Hausdorff compact space then we may add to this list:

$f$ is continuous;^[5]

and if both $X$ and $Y$ are first-countable spaces then we may add to this list:

$f$ has a sequentially closed graph (in $X \times Y$ );

Function with a sequentially closed graph

If $f : X \to Y$ is a function then the following are equivalent:

$f$ has a sequentially closed graph (in $X \times Y$ );
(definition) the graph of $f$ is a sequentially closed subset of $X \times Y$ ;
for every $x \in X$ and sequence $x • = (x i) \infty i =1$ in $X$ such that $x • \to x$ in $X$ , if $y \in Y$ is such that the net $f (x •) := (f (x i)) \infty i =1 \to y$ in $Y$ then $y = f (x)$ ;^[4]

set-valued function with a closed graph

If $F : X \to 2 Y$ is a set-valued function between topological spaces $X$ and $Y$ then the following are equivalent:

$F$ has a closed graph (in $X \times Y$ );
(definition) the graph of $F$ is a closed subset of $X \times Y$ ;

and if $Y$ is compact and Hausdorff then we may add to this list:

$F$ is upper hemicontinuous and $F (x)$ is a closed subset of $Y$ for all $x \in X$ ;^[6]

and if both $X$ and $Y$ are metrizable spaces then we may add to this list:

for all $x \in X$ , $y \in Y$ , and sequences $x • = (x i) \infty i =1$ in $X$ and $y • = (y i) \infty i =1$ in $Y$ such that $x • \to x$ in $X$ and $y • \to y$ in $Y$ , and $y i \in F (x i)$ for all $i$ , then $y \in F (x)$ .^{[citation needed]}

Characterizations of closed graphs (general topology)

Throughout, let $X$ and $Y$ be topological spaces and $X\times Y$ is endowed with the product topology.

Function with a closed graph

If $f:X\to Y$ is a function then it is said to have a closed graph if it satisfies any of the following are equivalent conditions:

(Definition): The graph $\operatorname {graph} f$ of $f$ is a closed subset of $X\times Y.$
For every $x\in X$ and net $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ in $X$ such that $x_{\bullet }\to x$ in $X,$ if $y\in Y$ is such that the net $f\left(x_{\bullet }\right)=\left(f\left(x_{i}\right)\right)_{i\in I}\to y$ in $Y$ then $y=f(x).$ ^[4]
- Compare this to the definition of continuity in terms of nets, which recall is the following: for every $x\in X$ and net $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ in $X$ such that $x_{\bullet }\to x$ in $X,$ $f\left(x_{\bullet }\right)\to f(x)$ in $Y.$
- Thus to show that the function $f$ has a closed graph, it may be assumed that $f\left(x_{\bullet }\right)$ converges in $Y$ to some $y\in Y$ (and then show that $y=f(x)$ ) while to show that $f$ is continuous, it may not be assumed that $f\left(x_{\bullet }\right)$ converges in $Y$ to some $y\in Y$ and instead, it must be proven that this is true (and moreover, it must more specifically be proven that $f\left(x_{\bullet }\right)$ converges to $f(x)$ in $Y$ ).

and if $Y$ is a Hausdorff compact space then we may add to this list:

$f$ is continuous.^[5]

and if both $X$ and $Y$ are first-countable spaces then we may add to this list:

$f$ has a sequentially closed graph in $X\times Y.$

Function with a sequentially closed graph

If $f:X\to Y$ is a function then the following are equivalent:

$f$ has a sequentially closed graph in $X\times Y.$
Definition: the graph of $f$ is a sequentially closed subset of $X\times Y.$
For every $x\in X$ and sequence $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ in $X$ such that $x_{\bullet }\to x$ in $X,$ if $y\in Y$ is such that the net $f\left(x_{\bullet }\right):=\left(f\left(x_{i}\right)\right)_{i=1}^{\infty }\to y$ in $Y$ then $y=f(x).$ ^[4]

Sufficient conditions for a closed graph

If f : X → Y is a continuous function between topological spaces and if Y is Hausdorff then f has a closed graph in X × Y.^[4]
- Note that if $f : X \to Y$ is a function between Hausdorff topological spaces then it is possible for $f$ to have a closed graph in $X \times Y$ but not be continuous.

Closed graph theorems: When a closed graph implies continuity

Conditions that guarantee that a function with a closed graph is necessarily continuous are called closed graph theorems. Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous.

If $f : X \to Y$ is a function between topological spaces whose graph is closed in $X \times Y$ and if $Y$ is a compact space then $f : X \to Y$ is continuous.^[4]

Examples

For examples in functional analysis, see continuous linear operator.

Continuous but not closed maps

Let $X$ denote the real numbers $ℝ$ with the usual Euclidean topology and let $Y$ denote $ℝ$ with the indiscrete topology (where note that $Y$ is not Hausdorff and that every function valued in $Y$ is continuous). Let $f : X \to Y$ be defined by $f (0) = 1$ and $f (x) = 0$ for all $x \neq 0$ . Then $f : X \to Y$ is continuous but its graph is not closed in $X \times Y$ .^[4]
If $X$ is any space then the identity map $Id : X \to X$ is continuous but its graph, which is the diagonal $Gr Id := { (x, x) : x \in X}$ , is closed in $X \times X$ if and only if $X$ is Hausdorff.^[7] In particular, if $X$ is not Hausdorff then $Id : X \to X$ is continuous but not closed.
If $f : X \to Y$ is a continuous map whose graph is not closed then $Y$ is not a Hausdorff space.

Closed but not continuous maps

Let $X$ and $Y$ both denote the real numbers $ℝ$ with the usual Euclidean topology. Let $f : X \to Y$ be defined by $f (0) = 0$ and $f(x) = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/x⁠$ for all $x \neq 0$ . Then $f : X \to Y$ has a closed graph (and a sequentially closed graph) in $X \times Y = ℝ 2$ but it is not continuous (since it has a discontinuity at $x = 0$ ).^[4]
Let $X$ denote the real numbers $ℝ$ with the usual Euclidean topology, let $Y$ denote $ℝ$ with the discrete topology, and let $Id : X \to Y$ be the identity map (i.e. $Id(x) := x$ for every $x \in X$ ). Then $Id : X \to Y$ is a linear map whose graph is closed in $X \times Y$ but it is clearly not continuous (since singleton sets are open in $Y$ but not in $X$ ).^[4]
Let $(X, 𝜏)$ be a Hausdorff TVS and let $𝜐$ be a vector topology on $X$ that is strictly finer than $𝜏$ . Then the identity map $Id : (X, 𝜏) \to (X, 𝜐)$ is a closed discontinuous linear operator.^[8]

References

^ Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.
^ Ursescu, Corneliu (1975). "Multifunctions with convex closed graph". Czechoslovak Mathematical Journal. 25 (3): 438–441. doi:10.21136/CMJ.1975.101337. ISSN 0011-4642.
^ Shafer, Wayne; Sonnenschein, Hugo (1975-12-01). "Equilibrium in abstract economies without ordered preferences" (PDF). Journal of Mathematical Economics. 2 (3): 345–348. doi:10.1016/0304-4068(75)90002-6. hdl:10419/220454. ISSN 0304-4068.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j Narici & Beckenstein 2011, pp. 459–483.
^ ^a ^b Munkres 2000, p. 171.
^ Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.
^ Rudin p.50
^ Narici & Beckenstein 2011, p. 480.

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[1] Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.

[2] Ursescu, Corneliu (1975). "Multifunctions with convex closed graph". Czechoslovak Mathematical Journal. 25 (3): 438–441. doi:10.21136/CMJ.1975.101337. ISSN 0011-4642.

[:2-3] Shafer, Wayne; Sonnenschein, Hugo (1975-12-01). "Equilibrium in abstract economies without ordered preferences" (PDF). Journal of Mathematical Economics. 2 (3): 345–348. doi:10.1016/0304-4068(75)90002-6. hdl:10419/220454. ISSN 0304-4068.

[FOOTNOTENariciBeckenstein2011459–483-4] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j Narici & Beckenstein 2011, pp. 459–483.

[FOOTNOTEMunkres2000171-5] Munkres 2000, p. 171.

[aliprantis-6] Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.

[7] Rudin p.50

[FOOTNOTENariciBeckenstein2011480-8] Narici & Beckenstein 2011, p. 480.

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