Closed graph property

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A real function y = f ( x ) {\displaystyle 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 → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × 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 → 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.

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"

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) 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:

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

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:

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

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

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

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.