Closed graph property

A discontinuous function with a closed graph: there are no "missing" (limit) points source ↗

Discontinuous function with a graph that is not closed: the point $(0,0)$ is a limit point that is not a point on the graph. source ↗

{\displaystyle (0,0)} — Discontinuous function with a graph that is not closed: the point $(0,0)$ is a limit point that is not a point on the graph. source ↗

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.¹² 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

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, $Gr f = Gr F$ .

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: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

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);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)

.

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 $x\in X$ $x\in X$ and net $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ in $X$ $X$ such that $x_{\bullet }\to x$ $x_{\bullet }\to x$ in $X,$ $X,$ if $y\in Y$ $y\in Y$ is such that the net $f\left(x_{\bullet }\right)=\left(f\left(x_{i}\right)\right)_{i\in I}\to y$ $f\left(x_{\bullet }\right)=\left(f\left(x_{i}\right)\right)_{i\in I}\to y$ in $Y$ $Y$ then $y=f(x).$ $y=f(x).$ ³
- Compare this to the definition of continuity in terms of nets, which recall is the following: for every $x\in X$ and net $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ 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.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 $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ 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 $y=f(x).$ 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.

Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis (PDF). Mathematical Surveys and Monographs. Vol. 53. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-0780-4. OCLC 37141279.
Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. (accessible to patrons with print disabilities)
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

[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.

[FOOTNOTENariciBeckenstein2011459–483-3] Narici & Beckenstein 2011, pp. 459–483.

[FOOTNOTEMunkres2000171-4] Munkres 2000, p. 171.

[aliprantis-5] Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.

[6] Rudin p.50

[FOOTNOTENariciBeckenstein2011480-7] Narici & Beckenstein 2011, p. 480.

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Definitions

Graphs and set-valued functions

Closed graph

Closable maps and closures

Closed maps and closed linear operators

Characterizations

Characterizations of closed graphs (general topology)

Function with a closed graph

Sufficient conditions for a closed graph

Closed graph theorems

Examples

Continuous but not closed maps

Closed but not continuous maps

See also

References