Closed graph property explained

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.^{[1] [2]} A function between topological spaces has a closed graph if its graph is a closed subset of the product space . 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 is the set

.

Notation: If is a set then the power set of, which is the set of all subsets of, is denoted by or .

Definition: If and are sets, a set-valued function in on (also called a -valued multifunction on) is a function with domain that is valued in . That is, is a function on such that for every, is a subset of .

• Some authors call a function a set-valued function only if it satisfies the additional requirement that is not empty for every ; this article does not require this.

Definition and notation: If is a set-valued function in a set then the graph of is the set

.

Definition: A function can be canonically identified with the set-valued function defined by for every, where is called the canonical set-valued function induced by (or associated with) .

• Note that in this case, .

Open and closed graph

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

Assumptions: Throughout, and are topological spaces,, and is a -valued function or set-valued function on (i.e. or). will always be endowed with the product topology.

Definition: We say that   has a closed graph (resp. open graph, sequentially closed graph, sequentially open graph) in if the graph of,, is a closed (resp. open, sequentially closed, sequentially open) subset of when is endowed with the product topology. If or if is clear from context then we may omit writing "in "

Observation: If is a function and is the canonical set-valued function induced by   (i.e. is defined by for every) then since, has a closed (resp. sequentially closed, open, sequentially open) graph in if and only if the same is true of .

Closable maps and closures

Definition: We say that the function (resp. set-valued function) is closable in if there exists a subset containing and a function (resp. set-valued function) whose graph is equal to the closure of the set in . Such an is called a closure of in , is denoted by, and necessarily extends .

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

Definition: If is closable on then a core or essential domain of is a subset such that the closure in of the graph of the restriction of to is equal to the closure of the graph of in (i.e. the closure of in is equal to the closure of in).

Closed maps and closed linear operators

Definition and notation: When we write then we mean that is a -valued function with domain where . If we say that is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) then we mean that the graph of is closed (resp. sequentially closed) in (rather than in).

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

Definition: A map is called closed if its graph is closed in . 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, " is closed" may instead mean the following:

Definition: A map between topological spaces is called a closed map if the image of a closed subset of is a closed subset of .

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 and be topological spaces.

Function with a closed graph

If is a function then the following are equivalent:

1.   has a closed graph (in);
2. (definition) the graph of,, is a closed subset of ;
3. for every and net in such that in, if is such that the net in then ;
   • Compare this to the definition of continuity in terms of nets, which recall is the following: for every and net in such that in, in .
   • Thus to show that the function has a closed graph we may assume that converges in to some (and then show that) while to show that is continuous we may not assume that converges in to some and we must instead prove that this is true (and moreover, we must more specifically prove that converges to in).

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

4.   is continuous;

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

5.   has a sequentially closed graph (in);

Function with a sequentially closed graph

If is a function then the following are equivalent:

1.   has a sequentially closed graph (in);
2. (definition) the graph of is a sequentially closed subset of ;
3. for every and sequence in such that in, if is such that the net in then ;

set-valued function with a closed graph

If is a set-valued function between topological spaces and then the following are equivalent:

1.   has a closed graph (in);
2. (definition) the graph of is a closed subset of ;

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

3. is upper hemicontinuous and is a closed subset of for all ;^[4]

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

4. for all,, and sequences in and in such that in and in, and for all, then .

Characterizations of closed graphs (general topology)

Throughout, let

X and Y be topological spaces and X x 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 if it satisfies any of the following are equivalent conditions:

1. (Definition): The graph \operatorname{graph} f of f is a closed subset of X x Y.
2. For every x \in X and net x _{\bull} = \left(x _{i\right)} _{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} \to 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 \in X and net x _{\bull} = \left(x _{i\right)} _{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:

3. f is continuous.

and if both

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

4. f has a sequentially closed graph in X x Y.

Function with a sequentially closed graph

If

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

1. f has a sequentially closed graph in X x Y.
2. Definition: the graph of f is a sequentially closed subset of X x Y.
3. For every x \in X and sequence x _{\bull} = \left(x _{i\right)} ^{inf ty} ^{i =1} 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)} ^{inf ty} ^{i =1} \to y in Y then y = f(x).

Sufficient conditions for a closed graph

• If is a continuous function between topological spaces and if is Hausdorff then   has a closed graph in .
  • Note that if is a function between Hausdorff topological spaces then it is possible for   to have a closed graph in but not be continuous.

Closed graph theorems: When a closed graph implies continuity

See main article: Closed graph theorem.

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 is a function between topological spaces whose graph is closed in and if is a compact space then is continuous.

Examples

For examples in functional analysis, see continuous linear operator.

Continuous but not closed maps

• Let denote the real numbers with the usual Euclidean topology and let denote with the indiscrete topology (where note that is not Hausdorff and that every function valued in is continuous). Let be defined by and for all . Then is continuous but its graph is not closed in .
• If is any space then the identity map is continuous but its graph, which is the diagonal, is closed in if and only if is Hausdorff.^[5] In particular, if is not Hausdorff then is continuous but not closed.
• If is a continuous map whose graph is not closed then is not a Hausdorff space.

Closed but not continuous maps

• Let and both denote the real numbers with the usual Euclidean topology. Let be defined by and for all . Then has a closed graph (and a sequentially closed graph) in but it is not continuous (since it has a discontinuity at).
• Let denote the real numbers with the usual Euclidean topology, let denote with the discrete topology, and let be the identity map (i.e. for every). Then is a linear map whose graph is closed in but it is clearly not continuous (since singleton sets are open in but not in).
• Let be a Hausdorff TVS and let be a vector topology on that is strictly finer than . Then the identity map is a closed discontinuous linear operator.

Notes and References

1. Baggs. Ivan. 1974. Functions with a closed graph. Proceedings of the American Mathematical Society. en. 43. 2. 439–442. 10.1090/S0002-9939-1974-0334132-8. 0002-9939. free.
2. Ursescu. Corneliu. 1975. Multifunctions with convex closed graph. Czechoslovak Mathematical Journal. 25. 3. 438–441. 10.21136/CMJ.1975.101337 . 0011-4642. free.
3. Shafer. Wayne. Sonnenschein. Hugo. 1975-12-01. Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics. 2. 3. 345–348. 10.1016/0304-4068(75)90002-6. 0304-4068. 10419/220454. free.
4. Book: Aliprantis, Charlambos. Infinite Dimensional Analysis: A Hitchhiker's Guide. Kim C. Border. Springer. 1999. 3rd. Chapter 17.
5. Rudin p.50