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{{Short description|Graph of a map closed in the product space}}
{{redirect|Open graph|the Facebook API|Open Graph Protocol}}
In [[mathematics]], particularly in [[functional analysis]] and [[topology]], '''closed graph''' is a property of [[Function (mathematics)|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}}</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}}</ref>
A function {{math|''f'' : ''X'' → ''Y''}} between [[topological space]]s has a '''closed graph''' if its [[Graph of a function |graph]] is a [[closed subset]] of the [[Product topology|product space]] {{math|''X'' × ''Y''}}.
A related property is '''open graph'''.<ref name=":2">{{Cite journal|last1=Shafer|first1=Wayne|last2=Sonnenschein|first2=Hugo|date=1975-12-01|title=Equilibrium in abstract economies without ordered preferences|journal=Journal of Mathematical Economics|volume=2|issue=3|pages=345–348|doi=10.1016/0304-4068(75)90002-6|issn=0304-4068|url=http://www.kellogg.northwestern.edu/research/math/papers/94.pdf|hdl=10419/220454|hdl-access=free}}</ref>

This property is studied because there are many theorems, known as [[closed graph theorem]]s, giving conditions under which a function with a closed graph is necessarily [[Continuous function (topology)|continuous]]. One particularly well-known class of closed graph theorems are the [[Closed graph theorem (functional analysis)|closed graph theorems in functional analysis]].

== Definitions ==

=== Graphs and set-valued functions ===

:'''Definition and notation''': The '''[[Graph of a function|graph]]''' of a [[Function (mathematics)|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 (function)|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''}}.

=== Open and 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 operator]]s 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 space]]s, {{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''' (resp. '''open graph''', '''sequentially closed graph''', '''sequentially open graph''') '''in {{math|''X'' × ''Y''}}''' if the graph of {{mvar|f}}, {{math|Gr ''f''}}, is a [[Closed set|closed]] (resp. [[Open set|open]], [[sequentially closed]], [[sequentially open]]) 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, open, sequentially open) 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 space]]s 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 space]]s (TVSs) (e.g. [[Banach space]]s) 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 '''[[Open and closed maps|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 (mathematics)|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|Hausdorff]] [[compact space]] 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 space|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 (mathematics)|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 space |compact]] and [[Hausdorff space|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>

== Sufficient conditions for a closed graph ==

* If {{math|''f'' : ''X'' → ''Y''}} is a [[continuous function]] between topological spaces and if {{mvar|Y}} is [[Hausdorff space|Hausdorff]] then {{mvar|f}}  has a closed graph in {{math|''X'' × ''Y''}}.{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}
** Note that if {{math|''f'' : ''X'' → ''Y''}} 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: When a closed graph implies continuity ==
{{Main|Closed graph theorem}}

Conditions that guarantee that a function with a closed graph is necessarily continuous are called '''[[closed graph theorem]]s'''.
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 ==

=== 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)}} a closed discontinuous linear operator.{{sfn | Narici | Beckenstein | 2011 | p=480}}

== Closed linear operators ==
{{See also|Unbounded operator#Closed linear operators}}

Every continuous linear operator valued in a Hausdorff [[topological vector space]] (TVS) has a closed graph and recall that a linear operator between two [[normed space]]s is continuous if and only if it is [[Bounded linear operator|bounded]].

:'''Definition''': If {{mvar|X}} and {{mvar|Y}} are [[topological vector space]]s (TVSs) then we call a [[linear map]] {{math|''f'' : ''D''(''f'') ⊆ ''X'' → ''Y''}} a '''closed linear operator''' if its graph is closed in {{math|''X'' × ''Y''}}.

=== Closed graph theorem ===

The [[closed graph theorem]] states that any closed linear operator {{math|''f'' : ''X'' → ''Y''}} between two [[F-space]]s (such as [[Banach space]]s) is continuous, where recall that if {{mvar|X}} and {{mvar|Y}} are [[Banach space]]s then {{math|''f'' : ''X'' → ''Y''}} being continuous is equivalent to {{mvar|f}} being bounded.

=== Basic properties ===

The following properties are easily checked for a linear operator {{math|''f'' : ''D''(''f'') ⊆ ''X'' → ''Y''}} between Banach spaces:

* If {{mvar|A}} is closed then {{math|''A'' − ''λ''Id''D''(''f'')}} is closed where {{mvar|λ}} is a scalar and {{math|Id''D''(''f'')}} is the [[identity function]];
* If {{mvar|f}} is closed, then its [[Kernel (linear operator)|kernel]] (or nullspace) is a closed vector subspace of {{mvar|X}};
* If {{mvar|f}} is closed and [[injective function|injective]] then its [[inverse function|inverse]] {{math|''f'' −1}} is also closed;
* A linear operator {{mvar|f}} admits a closure if and only if for every {{math|''x'' ∈ ''X''}} and every pair of sequences {{math|1=''x''• = (''x''''i''){{su|p=∞|b=''i''=1}}}} and {{math|1=''y''• = (''y''''i''){{su|p=∞|b=''i''=1}}}} in {{math|''D''(''f'')}} both converging to {{mvar|x}} in {{mvar|X}}, such that both {{math|1=''f''(''x''•) = (''f''(''x''''i'')){{su|p=∞|b=''i''=1}}}} and {{math|1=''f''(''y''•) = (''f''(''y''''i'')){{su|p=∞|b=''i''=1}}}} converge in {{mvar|Y}}, one has {{math|1=lim''i'' → ∞ ''fx''''i'' = lim''i'' → ∞ ''fy''''i''}}.

=== Example ===

Consider the [[derivative]] operator {{math|1=''A'' = {{sfrac|''d''|''dx''}}}} where {{math|1=''X'' = ''Y'' = ''C''([''a'', ''b''])}} is the Banach space of all [[continuous function]]s on an [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}}.
If one takes its domain {{math|''D''(''f'')}} to be {{math|''C''1([''a'', ''b''])}}, then {{mvar|f}} is a closed operator, which is not bounded.<ref>{{Cite book|title=Introductory Functional Analysis With Applications|last=Kreyszig|first=Erwin|publisher=John Wiley & Sons. Inc.|year=1978|isbn=0-471-50731-8|location=USA|pages=294}}</ref>
On the other hand if {{math|1=''D''(''f'') = [[smooth function{{!}}''C''∞([''a'', ''b''])]]}}, then {{mvar|f}} will no longer be closed, but it will be closable, with the closure being its extension defined on {{math|''C''1([''a'', ''b''])}}.

== 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}}

== References ==
{{reflist}}

* {{Köthe Topological Vector Spaces I}} 
* {{Kriegl Michor The Convenient Setting of Global Analysis}} 
* {{Munkres Topology|edition=2}} 
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} 
* {{Robertson Topological Vector Spaces}} 
* {{Rudin Walter Functional Analysis|edition=2}} 
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} 
* {{Swartz An Introduction to Functional Analysis}} 
* {{Trèves François Topological vector spaces, distributions and kernels}} 
* {{Wilansky Modern Methods in Topological Vector Spaces}} 

[[Category:Functional analysis]]

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