Editing Closed graph theorem (section)

== Closed graph theorem in point-set topology ==

In [[point-set topology]], the closed graph theorem states the following:

{{Math theorem
| name = Closed graph theorem{{sfn|Munkres|2000|pp=163–172}}
| math_statement = If <math>f : X \to Y</math> is a map from a [[topological space]] <math>X</math> into a [[Hausdorff space]] <math>Y,</math> then the graph of <math>f</math> is closed if <math>f : X \to Y</math> is [[Continuous function (topology)|continuous]]. The converse is true when <math>Y</math> is [[Compact space|compact]]. (Note that compactness and Hausdorffness do not imply each other.)
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{{Math proof|title=Proof|drop=hidden|proof=

First part: just note that the graph of <math>f</math> is the same as the pre-image <math>(f \times \operatorname{id}_Y)^{-1}(D)</math> where <math>D = \{ (y, y) \mid y \in Y \}</math> is the diagonal in <math>Y^2</math>.

Second part: 

For any open <math>V\subset Y</math> , we check <math>f^{-1}(V)</math> is open. So take any <math>x\in f^{-1}(V)</math> , we construct some open neighborhood <math>U</math> of <math>x</math> , such that <math>f(U)\subset V</math> .

Since the graph of <math>f</math> is closed, for every point <math>(x, y')</math> on the "vertical line at x", with <math>y'\neq f(x)</math> , draw an open rectangle <math>U_{y'}\times V_{y'}</math> disjoint from the graph of <math>f</math> . These open rectangles, when projected to the y-axis, cover the y-axis except at <math>f(x)</math> , so add one more set <math>V</math>.

Naively attempting to take <math>U:= \bigcap_{y'\neq f(x)} U_{y'}</math> would construct a set containing <math>x</math>, but it is not guaranteed to be open, so we use compactness here.

Since <math>Y</math> is compact, we can take a finite open covering of <math>Y</math> as <math>\{V, V_{y'_1}, ..., V_{y'_n}\}</math>.

Now take <math>U:= \bigcap_{i=1}^n U_{y'_i}</math>. It is an open neighborhood of <math>x</math>, since it is merely a finite intersection. We claim this is the open neighborhood <math>U</math> of <math>x</math> that we want.

Suppose not, then there is some unruly <math>x'\in U</math> such that <math>f(x') \not\in V</math> , then that would imply <math>f(x')\in V_{y'_i}</math> for some <math>i</math> by open covering, but then <math>(x', f(x'))\in U\times V_{y'_i} \subset U_{y'_i}\times V_{y'_i}</math> , a contradiction since it is supposed to be disjoint from the graph of <math>f</math> .
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If ''X'', ''Y'' are compact Hausdorff spaces, then the theorem can also be deduced from the open mapping theorem for such spaces; see {{section link||Relation_to_the_open_mapping_theorem}}.<!-- maybe the full version too? -->

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact <math>Y</math> is the real line, which allows the discontinuous function with closed graph <math>f(x) = \begin{cases}
\frac 1 x \text{ if }x\neq 0,\\
0\text{ else}
\end{cases}</math>.

Also, [[closed linear operator]]s in functional analysis (linear operators with closed graphs) are typically not continuous.

=== For set-valued functions ===

{{Math theorem
| name = Closed graph theorem for set-valued functions<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>
| math_statement = For a [[Hausdorff space|Hausdorff]] [[Compact space|compact]] range space <math>Y</math>, a set-valued function <math>F : X \to 2^Y</math> has a closed graph if and only if it is [[upper hemicontinuous]] and {{math|''F''(''x'')}} is a closed set for all <math>x \in X</math>.
}}