Closed graph property


In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A real function 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 between topological spaces has a closed graph if its graph is a closed subset of the product space.
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

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 .
This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis.

Closable maps and closures

Closed maps and closed linear operators

When reading literature in functional analysis, if is a linear map between topological vector spaces then " is closed" will almost always means the following:
Otherwise, especially in literature about point-set topology, " is closed" may instead mean the following:
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 ;
  2. 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 ;
  4. * 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.
  5. * Thus to show that the function has a closed graph we may assume that converges in to some 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 if is a Hausdorff space that is compact, then we may add to this list:
  •   is continuous;
    • and if both and are first-countable spaces then we may add to this list:
  •   has a sequentially closed graph ;
    • ;Function with a sequentially closed graph
    If is a function then the following are equivalent:
    1.   has a sequentially closed graph ;
    2. 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 ;
    2. the graph of is a closed subset of ;
    and if is compact and Hausdorff then we may add to this list:
  • is upper hemicontinuous and is a closed subset of for all ;
    • and if both and are metrizable spaces then we may add to this list:
  • for all,, and sequences in and in such that in and in, and for all, then.

    • Characterizations of closed graphs (general topology)

    Throughout, let and be topological spaces and is endowed with the product topology.

    Function with a closed graph

    If is a function then it is said to have a if it satisfies any of the following are equivalent conditions:

    1. : The graph of is a closed subset of
    2. 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, it may be assumed that converges in to some while to show that is continuous, it may not be assumed that converges in to some and instead, it must be proven that this is true.
    and if is a Hausdorff compact space then we may add to this list:

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

    1. 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

    Sufficient conditions for a closed graph

    • If is a continuous function between topological spaces and if is Hausdorff then   has a closed graph in. However, 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

    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

    Continuous but not closed maps

    • Let denote the real numbers with the usual Euclidean topology and let denote with the indiscrete topology. 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. 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 in but it is not continuous.
    • Let denote the real numbers with the usual Euclidean topology, let denote with the discrete topology, and let be the identity map. Then is a linear map whose graph is closed in but it is clearly not continuous.
    • 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.