Closed graph property

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.^[1]^[2] A real function y=f(x) is closed if the graph is closed, meaning that it contains all of its limit points. Every such continuous function has a closed graph, but the converse is not necessarily true.

More generally, a function $f : X \to Y$ between topological spaces has a closed graph if its graph is a closed subset of the product space $X \times Y$ .

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

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, $Gr f = Gr F$ .

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

f : S \to 2 Y

X \times Y

will always be endowed with the product topology.

Definition:^[3] We say that f has a closed graph in $X \times Y$ if the graph of f,

Gr f

, is a closed 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) 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

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

X \times Y

is equal to the closure of

Gr f | D

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);^[3]
- 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 space that is compact, then we may add to this list:

f is continuous;^[4]

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 $y = f (x)$ ;^[3]

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

;^[5]

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_{\bull} = \left(x_i\right)_{i \in I} in X such that x_{\bull} \to x in X, if y \in Y is such that the net f\left(x_{\bull}\right) = \left(f\left(x_i\right)\right)_{i \in I} \to y in Y then y = f(x).^[3]
- Compare this to the definition of continuity in terms of nets, which recall is the following: for every x \in X and net x_{\bull} = \left(x_i\right)_{i \in I} in X such that x_{\bull} \to x in X, f\left(x_{\bull}\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_{\bull}\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_{\bull}\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_{\bull}\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.^[4]

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_{\bull} = \left(x_i\right)_{i=1}^{\infty} in X such that x_{\bull} \to x in X, if y \in Y is such that the net f\left(x_{\bull}\right) := \left(f\left(x_i\right)\right)_{i=1}^{\infty} \to y in Y then y = f(x).^[3]

Sufficient conditions for a closed graph

If $f : X \to Y$ is a continuous function between topological spaces and if Y is Hausdorff then f has a closed graph in $X \times Y$ .^[3] However, if $f$ 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

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.^[3]

Examples

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 $X \times Y$ .^[3]
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.^[6] 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 $f (x) = 1 ⁄ x$ 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 $x = 0$ ).^[3]
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).^[3]
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.^[7]

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
^ Narici & Beckenstein 2011, pp. 459–483.
^ 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.