Editing Closed graph theorem (section)

== Graphs and maps with closed graphs ==
{{Main|Closed graph}}

If <math>f : X \to Y</math> is a map between [[topological space]]s then the '''graph''' of <math>f</math> is the set <math>\Gamma_f := \{ (x, f(x)) : x \in X \}</math> or equivalently,
<math display=block>\Gamma_f := \{ (x, y) \in X \times Y : y = f(x) \}</math>
It is said that '''the graph of <math>f</math> is closed''' if <math>\Gamma_f</math> is a [[closed set|closed subset]] of <math>X \times Y</math> (with the [[product topology]]).

Any continuous function into a [[Hausdorff space]] has a closed graph (see {{section link||Closed_graph_theorem_in_point-set_topology}})

Any linear map, <math>L : X \to Y,</math> between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) <math>L</math> is sequentially continuous in the sense of the product topology, then the map <math>L</math> is continuous and its graph, {{math|Gr ''L''}}, is necessarily closed. Conversely, if <math>L</math> is such a linear map with, in place of (1a), the graph of <math>L</math> is (1b) known to be closed in the Cartesian product space <math>X \times Y</math>, then <math>L</math> is continuous and therefore necessarily sequentially continuous.{{sfn|Rudin|1991|p=51-52}} 

=== Examples of continuous maps that do ''not'' have a closed graph ===

If <math>X</math> is any space then the identity map <math>\operatorname{Id} : X \to X</math> is continuous but its graph, which is the diagonal <math>\Gamma_{\operatorname{Id}} := \{ (x, x) : x \in X \},</math>, is closed in <math>X \times X</math> if and only if <math>X</math> is Hausdorff.{{sfn|Rudin|1991|p=50}} In particular, if <math>X</math> is not Hausdorff then <math>\operatorname{Id} : X \to X</math> is continuous but does {{em|not}} have a closed graph. 

Let <math>X</math> denote the real numbers <math>\R</math> with the usual [[Euclidean topology]] and let <math>Y</math> denote <math>\R</math> with the [[indiscrete topology]] (where note that <math>Y</math> is {{em|not}} Hausdorff and that every function valued in <math>Y</math> is continuous). Let <math>f : X \to Y</math> be defined by <math>f(0) = 1</math> and <math>f(x) = 0</math> for all <math>x \neq 0</math>. Then <math>f : X \to Y</math> is continuous but its graph is {{em|not}} closed in <math>X \times Y</math>.{{sfn|Narici|Beckenstein|2011|pp=459-483}}