• Source: Closed graph property

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.
A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y.
A related property is open graph.
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 → Y is the set
Gr f := { (x, f(x)) : x ∈ X } = { (x, y) ∈ X × 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 2Y 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 → 2Y with domain X that is valued in 2Y. That is, F is a function on X such that for every x ∈ X, F(x) is a subset of Y.
Some authors call a function F : X → 2Y a set-valued function only if it satisfies the additional requirement that F(x) is not empty for every x ∈ X; this article does not require this.
Definition and notation: If F : X → 2Y is a set-valued function in a set Y then the graph of F is the set
Gr F := { (x, y) ∈ X × Y : y ∈ F(x) }.
Definition: A function f : X → Y can be canonically identified with the set-valued function F : X → 2Y defined by F(x) := { f(x) } for every x ∈ 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 ⊆ X, and f is a Y-valued function or set-valued function on S (i.e. f : S → Y or f : S → 2Y). X × Y will always be endowed with the product topology.
Definition: We say that f  has a closed graph in X × Y if the graph of f, Gr f, is a closed subset of X × Y when X × Y is endowed with the product topology. If S = X or if X is clear from context then we may omit writing "in X × Y"
Note that we may define an open graph, a sequentially closed graph, and a sequentially open graph in similar ways.

Observation: If g : S → Y is a function and G is the canonical set-valued function induced by g  (i.e. G : S → 2Y is defined by G(s) := { g(s) } for every s ∈ S) then since Gr g = Gr G, g has a closed (resp. sequentially closed, open, sequentially open) graph in X × 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 × Y if there exists a subset D ⊆ X containing S and a function (resp. set-valued function) F : D → Y whose graph is equal to the closure of the set Gr f in X × Y. Such an F is called a closure of f in X × 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 → 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 ⊆ S such that the closure in X × Y of the graph of the restriction f |D : D → Y of f to D is equal to the closure of the graph of f in X × Y (i.e. the closure of Gr f in X × Y is equal to the closure of Gr f |D in X × Y).




= Closed maps and closed linear operators

=
Definition and notation: When we write f : D(f) ⊆ X → Y then we mean that f is a Y-valued function with domain D(f) where D(f) ⊆ X. If we say that f : D(f) ⊆ X → 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 × Y (rather than in D(f) × Y).
When reading literature in functional analysis, if f : X → 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 → Y is called closed if its graph is closed in X × 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 → 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 → Y is a function then the following are equivalent:

f  has a closed graph (in X × Y);
(definition) the graph of f, Gr f, is a closed subset of X × Y;
for every x ∈ X and net x• = (xi)i ∈ I in X such that x• → x in X, if y ∈ Y is such that the net f(x•) := (f(xi))i ∈ I → y in Y then y = f(x);
Compare this to the definition of continuity in terms of nets, which recall is the following: for every x ∈ X and net x• = (xi)i ∈ I in X such that x• → x in X, f(x•) → 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 ∈ 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 ∈ 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;

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 × Y);

Function with a sequentially closed graph

If f : X → Y is a function then the following are equivalent:
f  has a sequentially closed graph (in X × Y);
(definition) the graph of f is a sequentially closed subset of X × Y;
for every x ∈ X and sequence x• = (xi)∞i=1 in X such that x• → x in X, if y ∈ Y is such that the net f(x•) := (f(xi))∞i=1 → y in Y then y = f(x);

set-valued function with a closed graph

If F : X → 2Y is a set-valued function between topological spaces X and Y then the following are equivalent:
F  has a closed graph (in X × Y);
(definition) the graph of F is a closed subset of X × 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 ∈ X;

and if both X and Y are metrizable spaces then we may add to this list:
for all x ∈ X, y ∈ Y, and sequences x• = (xi)∞i=1 in X and y• = (yi)∞i=1 in Y such that x• → x in X and y• → y in Y, and yi ∈ F(xi) for all i, then y ∈ F(x).




= Characterizations of closed graphs (general topology)

=
Throughout, let



X


{\displaystyle X}

and



Y


{\displaystyle Y}

be topological spaces and



X
×
Y


{\displaystyle X\times Y}

is endowed with the product topology.



Function with a closed graph

If



f
:
X
→
Y


{\displaystyle 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



graph
⁡
f


{\displaystyle \operatorname {graph} f}

of



f


{\displaystyle f}

is a closed subset of



X
×
Y
.


{\displaystyle X\times Y.}


For every



x
∈
X


{\displaystyle x\in X}

and net




x

∙


=


(

x

i


)


i
∈
I




{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}

in



X


{\displaystyle X}

such that




x

∙


→
x


{\displaystyle x_{\bullet }\to x}

in



X
,


{\displaystyle X,}

if



y
∈
Y


{\displaystyle y\in Y}

is such that the net



f

(

x

∙


)

=


(

f

(

x

i


)


)


i
∈
I


→
y


{\displaystyle f\left(x_{\bullet }\right)=\left(f\left(x_{i}\right)\right)_{i\in I}\to y}

in



Y


{\displaystyle Y}

then



y
=
f
(
x
)
.


{\displaystyle y=f(x).}


Compare this to the definition of continuity in terms of nets, which recall is the following: for every



x
∈
X


{\displaystyle x\in X}

and net




x

∙


=


(

x

i


)


i
∈
I




{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}

in



X


{\displaystyle X}

such that




x

∙


→
x


{\displaystyle x_{\bullet }\to x}

in



X
,


{\displaystyle X,}





f

(

x

∙


)

→
f
(
x
)


{\displaystyle f\left(x_{\bullet }\right)\to f(x)}

in



Y
.


{\displaystyle Y.}


Thus to show that the function



f


{\displaystyle f}

has a closed graph, it may be assumed that



f

(

x

∙


)



{\displaystyle f\left(x_{\bullet }\right)}

converges in



Y


{\displaystyle Y}

to some



y
∈
Y


{\displaystyle y\in Y}

(and then show that



y
=
f
(
x
)


{\displaystyle y=f(x)}

) while to show that



f


{\displaystyle f}

is continuous, it may not be assumed that



f

(

x

∙


)



{\displaystyle f\left(x_{\bullet }\right)}

converges in



Y


{\displaystyle Y}

to some



y
∈
Y


{\displaystyle y\in Y}

and instead, it must be proven that this is true (and moreover, it must more specifically be proven that



f

(

x

∙


)



{\displaystyle f\left(x_{\bullet }\right)}

converges to



f
(
x
)


{\displaystyle f(x)}

in



Y


{\displaystyle Y}

).

and if



Y


{\displaystyle Y}

is a Hausdorff compact space then we may add to this list:




f


{\displaystyle f}

is continuous.

and if both



X


{\displaystyle X}

and



Y


{\displaystyle Y}

are first-countable spaces then we may add to this list:




f


{\displaystyle f}

has a sequentially closed graph in



X
×
Y
.


{\displaystyle X\times Y.}


Function with a sequentially closed graph
If



f
:
X
→
Y


{\displaystyle f:X\to Y}

is a function then the following are equivalent:




f


{\displaystyle f}

has a sequentially closed graph in



X
×
Y
.


{\displaystyle X\times Y.}


Definition: the graph of



f


{\displaystyle f}

is a sequentially closed subset of



X
×
Y
.


{\displaystyle X\times Y.}


For every



x
∈
X


{\displaystyle x\in X}

and sequence




x

∙


=


(

x

i


)


i
=
1


∞




{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}

in



X


{\displaystyle X}

such that




x

∙


→
x


{\displaystyle x_{\bullet }\to x}

in



X
,


{\displaystyle X,}

if



y
∈
Y


{\displaystyle y\in Y}

is such that the net



f

(

x

∙


)

:=


(

f

(

x

i


)


)


i
=
1


∞


→
y


{\displaystyle f\left(x_{\bullet }\right):=\left(f\left(x_{i}\right)\right)_{i=1}^{\infty }\to y}

in



Y


{\displaystyle Y}

then



y
=
f
(
x
)
.


{\displaystyle y=f(x).}





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. However, if f is a function between Hausdorff topological spaces, then it is possible for f  to have a closed graph in X × 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 → Y is a function between topological spaces whose graph is closed in X × Y and if Y is a compact space then f : X → Y is continuous.




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 → Y be defined by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f : X → Y is continuous but its graph is not closed in X × Y.
If X is any space then the identity map Id : X → X is continuous but its graph, which is the diagonal Gr Id := { (x, x) : x ∈ X }, is closed in X × X if and only if X is Hausdorff. In particular, if X is not Hausdorff then Id : X → X is continuous but not closed.
If f : X → 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 → Y be defined by f(0) = 0 and f(x) = ⁠1/x⁠ for all x ≠ 0. Then f : X → Y has a closed graph (and a sequentially closed graph) in X × Y = ℝ2 but it is not continuous (since it has a discontinuity at x = 0).
Let X denote the real numbers ℝ with the usual Euclidean topology, let Y denote ℝ with the discrete topology, and let Id : X → Y be the identity map (i.e. Id(x) := x for every x ∈ X). Then Id : X → Y is a linear map whose graph is closed in X × Y but it is clearly not continuous (since singleton sets are open in Y but not in X).
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) → (X, 𝜐) is a closed discontinuous linear operator.




See also


Almost open linear map – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets
Closed graph theorem – Theorem relating continuity to graphs
Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
Kakutani fixed-point theorem – Fixed-point theorem for set-valued functions
Open mapping theorem (functional analysis) – Condition for a linear operator to be open
Webbed space – Space where open mapping and closed graph theorems hold




References



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 (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
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.

Kata Kunci Pencarian:

• Closed graph property
• Graph property
• Closed linear operator
• Property graph
• Closed graph theorem (functional analysis)
• Glossary of graph theory
• Graph minor
• Cycle (graph theory)
• Graph Query Language
• Cycle graph