Closed graph property

{{Short description|Property of functions in topology}} {{cleanup|reason=Need to remove a lot of duplication.|date=July 2024}} [[File:Reciprocal of x times characteristic function of non-positive reals.pdf|thumb|A discontinuous function with a closed graph: there are no "missing" (limit) points]] [[File:Characteristic function of non-negative reals.pdf|thumb|Discontinuous function with a graph that is not closed: the point <math>(0,0)</math> is a limit point that is not a point on the graph.]] In mathematics, particularly in functional analysis and topology, '''closed graph''' is a property of functions.<ref>{{Cite journal|last=Baggs|first=Ivan|date=1974|title=Functions with a closed graph|url=https://www.ams.org/|journal=Proceedings of the American Mathematical Society|language=en|volume=43|issue=2|pages=439–442|doi=10.1090/S0002-9939-1974-0334132-8|issn=0002-9939|doi-access=free|url-access=subscription}}</ref><ref>{{Cite journal|last=Ursescu|first=Corneliu|date=1975|title=Multifunctions with convex closed graph|url=https://eudml.org/doc/12881|journal=Czechoslovak Mathematical Journal|volume=25|issue=3|pages=438–441|doi=10.21136/CMJ.1975.101337 |issn=0011-4642|doi-access=free}}</ref> A real function <math>y=f(x)</math> 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 {{math|''f'' : ''X'' → ''Y''}} between topological spaces has a '''closed graph''' if its graph is a closed subset of the product space {{math|''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 {{math|''f'' : ''X'' → ''Y''}} is the set ::{{math|1=Gr ''f'' := { (''x'', ''f''(''x'')) : ''x'' ∈ ''X''} = { (''x'', ''y'') ∈ ''X'' × ''Y'' : ''y'' = ''f''(''x'')}<nowiki/>}}.

:'''Notation''': If {{mvar|Y}} is a set then the power set of {{mvar|Y}}, which is the set of all subsets of {{mvar|Y}}, is denoted by {{math|2^''Y''}} or {{math|𝒫(''Y'')}}.

:'''Definition''': If {{mvar|X}} and {{mvar|Y}} are sets, a '''set-valued function''' in {{mvar|Y}} on {{mvar|X}} (also called a {{mvar|Y}}-valued '''multifunction''' on {{mvar|X}}) is a function {{math|''F'' : ''X'' → 2^''Y''}} with domain {{mvar|X}} that is valued in {{math|2^''Y''}}. That is, {{mvar|F}} is a function on {{mvar|X}} such that for every {{math|''x'' ∈ ''X''}}, {{math|''F''(''x'')}} is a subset of {{mvar|Y}}. :* Some authors call a function {{math|''F'' : ''X'' → 2^''Y''}} a set-valued function only if it satisfies the additional requirement that {{math|''F''(''x'')}} is not empty for every {{math|''x'' ∈ ''X''}}; this article does not require this.

:'''Definition and notation''': If {{math|''F'' : ''X'' → 2^''Y''}} is a set-valued function in a set {{mvar|Y}} then the '''graph''' of {{mvar|F}} is the set ::{{math|1=Gr ''F'' := { (''x'', ''y'') ∈ ''X'' × ''Y'' : ''y'' ∈ ''F''(''x'')}<nowiki/>}}.

:'''Definition''': A function {{math|''f'' : ''X'' → ''Y''}} can be canonically identified with the set-valued function {{math|''F'' : ''X'' → 2^''Y''}} defined by {{math|1=''F''(''x'') := { ''f''(''x'')} }} for every {{math|''x'' ∈ ''X''}}, where {{mvar|F}} is called the '''canonical set-valued function''' induced by (or associated with) {{mvar|f}}. :*Note that in this case, {{math|1=Gr ''f'' = Gr ''F''}}.

=== Closed graph ===

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

:'''Assumptions''': Throughout, {{mvar|X}} and {{mvar|Y}} are topological spaces, {{math|''S'' ⊆ ''X''}}, and {{mvar|f}} is a {{mvar|Y}}-valued function or set-valued function on {{mvar|S}} (i.e. {{math|''f'' : ''S'' → ''Y''}} or {{math|''f'' : ''S'' → 2^''Y''}}). {{math|''X'' × ''Y''}} will always be endowed with the product topology.

:'''Definition''':{{sfn | Narici | Beckenstein | 2011 | pp=459-483}} We say that {{mvar|f}} has a '''closed graph''' '''in {{math|''X'' × ''Y''}}''' if the graph of {{mvar|f}}, {{math|Gr ''f''}}, is a closed subset of {{math|''X'' × ''Y''}} when {{math|''X'' × ''Y''}} is endowed with the product topology. If {{math|1=''S'' = ''X''}} or if {{mvar|X}} is clear from context then we may omit writing "in {{math|''X'' × ''Y''}}"

:'''Observation''': If {{math|''g'' : ''S'' → ''Y''}} is a function and {{mvar|G}} is the canonical set-valued function induced by {{mvar|g}} (i.e. {{math|''G'' : ''S'' → 2^''Y''}} is defined by {{math|1=''G''(''s'') := { ''g''(''s'')} }} for every {{math|''s'' ∈ ''S''}}) then since {{math|1=Gr ''g'' = Gr ''G''}}, {{mvar|g}} has a closed (resp. sequentially closed) graph in {{math|''X'' × ''Y''}} if and only if the same is true of {{mvar|G}}.

=== Closable maps and closures ===

:'''Definition''': We say that the function (resp. set-valued function) {{mvar|f}} is '''closable in {{math|''X'' × ''Y''}}''' if there exists a subset {{math|''D'' ⊆ ''X''}} containing {{mvar|S}} and a function (resp. set-valued function) {{math|''F'' : ''D'' → ''Y''}} whose graph is equal to the closure of the set {{math|Gr ''f''}} in {{math|''X'' × ''Y''}}. Such an {{mvar|F}} is called a '''closure of {{mvar|f}} in {{math|''X'' × ''Y''}}''', is denoted by {{math|{{overline|f}}}}, and necessarily extends {{mvar|f}}. :*'''Additional assumptions for linear maps''': If in addition, {{mvar|S}}, {{mvar|X}}, and {{mvar|Y}} are topological vector spaces and {{math|''f'' : ''S'' → ''Y''}} is a linear map then to call {{mvar|f}} closable we also require that the set {{mvar|D}} be a vector subspace of {{mvar|X}} and the closure of {{mvar|f}} be a linear map.

:'''Definition''': If {{mvar|f}} is closable on {{mvar|S}} then a '''core''' or '''essential domain''' of {{mvar|f}} is a subset {{math|''D'' ⊆ ''S''}} such that the closure in {{math|''X'' × ''Y''}} of the graph of the restriction {{math|''f'' {{big|{{!}}}}_''D'' : ''D'' → ''Y''}} of {{mvar|f}} to {{mvar|D}} is equal to the closure of the graph of {{mvar|f}} in {{math|''X'' × ''Y''}} (i.e. the closure of {{math|Gr ''f''}} in {{math|''X'' × ''Y''}} is equal to the closure of {{math|Gr ''f'' {{big|{{!}}}}_''D''}} in {{math|''X'' × ''Y''}}).

=== Closed maps and closed linear operators ===

:'''Definition and notation''': When we write {{math|''f'' : ''D''(''f'') ⊆ ''X'' → ''Y''}} then we mean that {{mvar|f}} is a {{mvar|Y}}-valued function with domain {{math|''D''(''f'')}} where {{math|''D''(''f'') ⊆ ''X''}}. If we say that {{math|''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 {{mvar|f}} is closed (resp. sequentially closed) in {{math|''X'' × ''Y''}} (rather than in {{math|''D''(''f'') × ''Y''}}).

When reading literature in functional analysis, if {{math|''f'' : ''X'' → ''Y''}} is a linear map between topological vector spaces (TVSs) (e.g. Banach spaces) then "{{mvar|f}} is closed" will almost always means the following:

:'''Definition''': A map {{math|''f'' : ''X'' → ''Y''}} is called '''closed''' if its graph is closed in {{math|''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, "{{mvar|f}} is closed" may instead mean the following:

:'''Definition''': A map {{math|''f'' : ''X'' → ''Y''}} between topological spaces is called a '''closed map''' if the image of a closed subset of {{mvar|X}} is a closed subset of {{mvar|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 {{mvar|X}} and {{mvar|Y}} be topological spaces.

;Function with a closed graph

If {{math|''f'' : ''X'' → ''Y''}} is a function then the following are equivalent:

# {{mvar|f}} has a closed graph (in {{math|''X'' × ''Y''}}); # (definition) the graph of {{mvar|f}}, {{math|Gr ''f''}}, is a closed subset of {{math|''X'' × ''Y''}}; # for every {{math|''x'' ∈ ''X''}} and net {{math|1=''x''_• = (''x''_''i'')_{''i'' ∈ ''I''}}} in {{mvar|X}} such that {{math|''x''_• → ''x''}} in {{mvar|X}}, if {{math|''y'' ∈ ''Y''}} is such that the net {{math|1=''f''(''x''_•) := (''f''(''x''_''i''))_{''i'' ∈ ''I''} → ''y''}} in {{mvar|Y}} then {{math|1=''y'' = ''f''(''x'')}};{{sfn | Narici | Beckenstein | 2011 | pp=459-483}} #* Compare this to the definition of continuity in terms of nets, which recall is the following: for every {{math|''x'' ∈ ''X''}} and net {{math|1=''x''_• = (''x''_''i'')_{''i'' ∈ ''I''}}} in {{mvar|X}} such that {{math|''x''_• → ''x''}} in {{mvar|X}}, {{math|''f''(''x''_•) → ''f''(''x'')}} in {{mvar|Y}}. #* Thus to show that the function {{mvar|f}} has a closed graph we ''may'' assume that {{math|''f''(''x''_•)}} converges in {{mvar|Y}} to some {{math|''y'' ∈ ''Y''}} (and then show that {{math|1=''y'' = ''f''(''x'')}}) while to show that {{mvar|f}} is continuous we may ''not'' assume that {{math|''f''(''x''_•)}} converges in {{mvar|Y}} to some {{math|''y'' ∈ ''Y''}} and we must instead prove that this is true (and moreover, we must more specifically prove that {{math|''f''(''x''_•)}} converges to {{math|''f''(''x'')}} in {{mvar|Y}}).

and if {{mvar|Y}} is a Hausdorff space that is compact, then we may add to this list: <li value="4">{{mvar|f}} is continuous;{{sfn | Munkres | 2000 | p=171}}</li>

and if both {{mvar|X}} and {{mvar|Y}} are first-countable spaces then we may add to this list: <li value="5">{{mvar|f}} has a sequentially closed graph (in {{math|''X'' × ''Y''}});</li>

;Function with a sequentially closed graph

If {{math|''f'' : ''X'' → ''Y''}} is a function then the following are equivalent: # {{mvar|f}} has a sequentially closed graph (in {{math|''X'' × ''Y''}}); # (definition) the graph of {{mvar|f}} is a sequentially closed subset of {{math|''X'' × ''Y''}}; # for every {{math|''x'' ∈ ''X''}} and sequence {{math|1=''x''_• = (''x''_''i''){{su|p=∞|b=''i''=1}}}} in {{mvar|X}} such that {{math|''x''_• → ''x''}} in {{mvar|X}}, if {{math|''y'' ∈ ''Y''}} is such that the net {{math|1=''f''(''x''_•) := (''f''(''x''_''i'')){{su|p=∞|b=''i''=1}} → ''y''}} in {{mvar|Y}} then {{math|1=''y'' = ''f''(''x'')}};{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}

;set-valued function with a closed graph

If {{math|''F'' : ''X'' → 2^''Y''}} is a set-valued function between topological spaces {{mvar|X}} and {{mvar|Y}} then the following are equivalent: # {{mvar|F}} has a closed graph (in {{math|''X'' × ''Y''}}); # (definition) the graph of {{mvar|F}} is a closed subset of {{math|''X'' × ''Y''}};

and if {{mvar|Y}} is compact and Hausdorff then we may add to this list:

<li value="3">{{mvar|F}} is upper hemicontinuous and {{math|''F''(''x'')}} is a closed subset of {{mvar|Y}} for all {{math|''x'' ∈ ''X''}};<ref name="aliprantis">{{cite book|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|last=Aliprantis|first=Charlambos|author2=Kim C. Border|publisher=Springer|year=1999|edition=3rd|chapter=Chapter 17}}</ref></li>

and if both {{mvar|X}} and {{mvar|Y}} are metrizable spaces then we may add to this list: <li value="4">for all {{math|''x'' ∈ ''X''}}, {{math|''y'' ∈ ''Y''}}, and sequences {{math|1=''x''_• = (''x''_''i''){{su|p=∞|b=''i''=1}}}} in {{mvar|X}} and {{math|1=''y''_• = (''y''_''i''){{su|p=∞|b=''i''=1}}}} in {{mvar|Y}} such that {{math|''x''_• → ''x''}} in {{mvar|X}} and {{math|''y''_• → ''y''}} in {{mvar|Y}}, and {{math|''y''_''i'' ∈ ''F''(''x''_''i'')}} for all {{mvar|i}}, then {{math|''y'' ∈ ''F''(''x'')}}.{{citation needed|date=August 2020}}</li>

===Characterizations of closed graphs (general topology)===

Throughout, let <math>X</math> and <math>Y</math> be topological spaces and <math>X \times Y</math> is endowed with the product topology.

====Function with a closed graph====

If <math>f : X \to Y</math> is a function then it is said to have a '''{{em|closed graph}}''' if it satisfies any of the following are equivalent conditions: <ol> <li>(Definition): The graph <math>\operatorname{graph} f</math> of <math>f</math> is a closed subset of <math>X \times Y.</math></li> <li>For every <math>x \in X</math> and net <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in <math>X</math> such that <math>x_{\bull} \to x</math> in <math>X,</math> if <math>y \in Y</math> is such that the net <math>f\left(x_{\bull}\right) = \left(f\left(x_i\right)\right)_{i \in I} \to y</math> in <math>Y</math> then <math>y = f(x).</math>{{sfn|Narici|Beckenstein|2011|pp=459-483}} * Compare this to the definition of continuity in terms of nets, which recall is the following: for every <math>x \in X</math> and net <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in <math>X</math> such that <math>x_{\bull} \to x</math> in <math>X,</math> <math>f\left(x_{\bull}\right) \to f(x)</math> in <math>Y.</math> * Thus to show that the function <math>f</math> has a closed graph, it ''may'' be assumed that <math>f\left(x_{\bull}\right)</math> converges in <math>Y</math> to some <math>y \in Y</math> (and then show that <math>y = f(x)</math>) while to show that <math>f</math> is continuous, it may ''not'' be assumed that <math>f\left(x_{\bull}\right)</math> converges in <math>Y</math> to some <math>y \in Y</math> and instead, it must be proven that this is true (and moreover, it must more specifically be proven that <math>f\left(x_{\bull}\right)</math> converges to <math>f(x)</math> in <math>Y</math>).</li> </ol>

and if <math>Y</math> is a Hausdorff compact space then we may add to this list: <ol start=3> <li><math>f</math> is continuous.{{sfn|Munkres|2000|p=171}}</li> </ol>

and if both <math>X</math> and <math>Y</math> are first-countable spaces then we may add to this list: <ol start=4> <li><math>f</math> has a sequentially closed graph in <math>X \times Y.</math></li> </ol>

'''Function with a sequentially closed graph'''

If <math>f : X \to Y</math> is a function then the following are equivalent: <ol> <li><math>f</math> has a sequentially closed graph in <math>X \times Y.</math></li> <li>Definition: the graph of <math>f</math> is a sequentially closed subset of <math>X \times Y.</math></li> <li>For every <math>x \in X</math> and sequence <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> in <math>X</math> such that <math>x_{\bull} \to x</math> in <math>X,</math> if <math>y \in Y</math> is such that the net <math>f\left(x_{\bull}\right) := \left(f\left(x_i\right)\right)_{i=1}^{\infty} \to y</math> in <math>Y</math> then <math>y = f(x).</math>{{sfn|Narici|Beckenstein|2011|pp=459-483}}</li> </ol>

== Sufficient conditions for a closed graph ==

* If {{math|''f'' : ''X'' → ''Y''}} is a continuous function between topological spaces and if {{mvar|Y}} is Hausdorff then {{mvar|f}} has a closed graph in {{math|''X'' × ''Y''}}.{{sfn | Narici | Beckenstein | 2011 | pp=459-483}} However, if {{math|''f''}} is a function between Hausdorff topological spaces, then it is possible for {{mvar|f}} to have a closed graph in {{math|''X'' × ''Y''}} but ''not'' be continuous.

== Closed graph theorems == {{Main|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 {{math|''f'' : ''X'' → ''Y''}} is a function between topological spaces whose graph is closed in {{math|''X'' × ''Y''}} and if {{mvar|Y}} is a compact space then {{math|''f'' : ''X'' → ''Y''}} is continuous.{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}

== Examples == {{Hatnote|For examples in functional analysis, see continuous linear operator}}

=== Continuous but not closed maps ===

* Let {{mvar|X}} denote the real numbers {{math|ℝ}} with the usual Euclidean topology and let {{mvar|Y}} denote {{math|ℝ}} with the indiscrete topology (where note that {{mvar|Y}} is ''not'' Hausdorff and that every function valued in {{mvar|Y}} is continuous). Let {{math|''f'' : ''X'' → ''Y''}} be defined by {{math|1=''f''(0) = 1}} and {{math|1=''f''(''x'') = 0}} for all {{math|''x'' ≠ 0}}. Then {{math|''f'' : ''X'' → ''Y''}} is continuous but its graph is ''not'' closed in {{math|''X'' × ''Y''}}.{{sfn | Narici | Beckenstein | 2011 | pp=459-483}} * If {{mvar|X}} is any space then the identity map {{math|Id : ''X'' → ''X''}} is continuous but its graph, which is the diagonal {{math|1=Gr Id := { (''x'', ''x'') : ''x'' ∈ ''X''}<nowiki/>}}, is closed in {{math|''X'' × ''X''}} if and only if {{mvar|X}} is Hausdorff.<ref>Rudin p.50</ref> In particular, if {{mvar|X}} is not Hausdorff then {{math|Id : ''X'' → ''X''}} is continuous but ''not'' closed. * If {{math|''f'' : ''X'' → ''Y''}} is a continuous map whose graph is not closed then {{mvar|Y}} is ''not'' a Hausdorff space.

=== Closed but not continuous maps ===

* Let {{mvar|X}} and {{mvar|Y}} both denote the real numbers {{math|ℝ}} with the usual Euclidean topology. Let {{math|''f'' : ''X'' → ''Y''}} be defined by {{math|1=''f''(0) = 0}} and {{math|1=''f''(''x'') = {{sfrac|1|''x''}}}} for all {{math|''x'' ≠ 0}}. Then {{math|''f'' : ''X'' → ''Y''}} has a closed graph (and a sequentially closed graph) in {{math|1=''X'' × ''Y'' = ℝ²}} but it is ''not'' continuous (since it has a discontinuity at {{math|1=''x'' = 0}}).{{sfn | Narici | Beckenstein | 2011 | pp=459-483}} * Let {{mvar|X}} denote the real numbers {{math|ℝ}} with the usual Euclidean topology, let {{mvar|Y}} denote {{math|ℝ}} with the discrete topology, and let {{math|Id : ''X'' → ''Y''}} be the identity map (i.e. {{math|1=Id(''x'') := ''x''}} for every {{math|''x'' ∈ ''X''}}). Then {{math|Id : ''X'' → ''Y''}} is a linear map whose graph is closed in {{math|1=''X'' × ''Y''}} but it is clearly ''not'' continuous (since singleton sets are open in {{mvar|Y}} but not in {{mvar|X}}).{{sfn | Narici | Beckenstein | 2011 | pp=459-483}} * Let {{math|(''X'', 𝜏)}} be a Hausdorff TVS and let {{math|𝜐}} be a vector topology on {{mvar|X}} that is strictly finer than {{math|𝜏}}. Then the identity map {{math|Id : (''X'', 𝜏) → (''X'', 𝜐)}} is a closed discontinuous linear operator.{{sfn | Narici | Beckenstein | 2011 | p=480}}

== See also ==

* {{annotated link|Almost open linear map}} * {{annotated link|Closed graph theorem}} * {{annotated link|Closed graph theorem (functional analysis)}} * {{annotated link|Kakutani fixed-point theorem}} * {{annotated link|Open mapping theorem (functional analysis)}} * {{annotated link|Webbed space}} * Graph continuous function

== References == {{reflist}}

* {{Köthe Topological Vector Spaces I}} <!-- {{sfn | Köthe | 1969 | p=}} --> * {{Kriegl Michor The Convenient Setting of Global Analysis}} <!-- {{sfn | Kriegl | 1997 | p=}} --> * {{Munkres Topology|edition=2}} <!-- {{sfn | Munkres | 2000 | p=}} --> * {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} --> * {{Robertson Topological Vector Spaces}} <!-- {{sfn | Robertson | 1964 | p=}} --> * {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} --> * {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} --> * {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn | Swartz | 1992 | p=}} --> * {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} --> * {{Wilansky Modern Methods in Topological Vector Spaces}} <!-- {{sfn | Wilansky | 2013 | p=}} -->

Category:Functional analysis