Could someone be so kind to point me in the direction of a citeable proof of the following version of the Closed Graph Theorem? (i.e. assuming this is true, could someone give me a literature reference, or if it is false, a counter-example.))


Suppose $A$ and $B$ are topological spaces. For the correspondence $f : A \to B$ with closed values (i.e. $f(a)$ - closed for all $a \in A$ and $f(a)\neq\emptyset$), closed domain and compact range, to be upper hemicontinuous it is sufficient and necessary for $f$ to have closed graph. 


Edited to add: $B$ is *not* assumed Hausdorff.



Reference or counter-example for Closed Graph Theorem for multivalued maps in general topological spaces
Since this result is mentioned in [Wikipedia article on hemicontinuity](https://en.wikipedia.org/wiki/Hemicontinuity#Closed_Graph_Theorem), the [references](https://en.wikipedia.org/wiki/Hemicontinuity#References) mentioned there might be a reasonable place to look for this result.


If you simply try searching for ["upper hemicontinuous" "closed graph"](https://www.google.com/search?q=%22upper+hemicontinuous%22+%22closed+graph%22&tbm=bks) in Google Books, there are several reasonably looking results. (Of course, you can also try [Google](https://www.google.com/search?q=%22upper+hemicontinuous%22+%22closed+graph%22) and [Google Scholar](https://scholar.google.com/scholar?q=%22upper+hemicontinuous%22+%22closed+graph%22).) Notice that some authors use the name \*upper semicontinuous instead of upper hemicontinuous. This gives us some other reasonable search phrases - again you can try [Google Books](https://www.google.com/search?q=%22upper+semicontinuous%22+%22closed+graph%22&tbm=bks), [Google](https://www.google.com/search?q=%22upper+semicontinuous%22+%22closed+graph%22) or [Google Scholar](https://scholar.google.com/scholar?q=%22upper+semicontinuous%22+%22closed+graph%22).




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Aliprantis C.D., Border K.C. *Infinite-dimensional analysis* (3ed., Springer, 2005), [page 561](https://books.google.com/books?id=4hIq6ExH7NoC&pg=PA561):



> 
> **17.9 Definition** A correspondence $\varphi \colon X \twoheadrightarrow Y$ between topological spaces is
>  **closed**, or **has closed graph**, if its graph 
>  $$\operatorname{Gr}\varphi = \{(x,y)\in X\times Y; y\in\varphi(x)\}$$
>  is a closed subset of $X\times Y$. 
> 
> 
> **17.11 Closed Graph Theorem** A correspondence with compact Hausdorff range space is closed if and only if it is upper hemicontinuous and closed-valued. 
> 
> 
> 




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If $\varphi$ is single-valued (i.e., $|\varphi(x)|=1$ for each $x$), then the definition of upper hemicontinuity becomes continuity and the above result says that a function with compact Hausdorff codomain is continuous if and only if it has closed graph. (This is also sometimes called [closed graph theorem](https://en.wikipedia.org/wiki/Closed_graph_theorem).) As can be seen from counterexamples given [here](https://math.stackexchange.com/questions/108709/is-the-graph-g-f-x-fx-in-x-times-y-x-in-x-a-closed-subset-of-x/108721#108721), the assumption that $Y$ is Hausdorff cannot be omitted in the closed graph theorem for functions. (There is an example of a function to a compact space, which is continuous and does not have closed graph.) Of course, this means that it cannot be omitted from the close graph theorem for upper hemicontinuous multifunctions, either.




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**EDIT:** Since I made this post, the Wikipedia article linked above was edited and now ([link to the current revision](https://en.wikipedia.org/w/index.php?title=Hemicontinuity&oldid=730453869)) it contains the following reference for the theorem in question:
J.-P. Aubin, H. Frankowska: *Set-Valued Analysis* (Birkhäuser, 1990), [page 42](https://books.google.com/books?id=cVUl3smcGlcC&pg=PA42)



> 
> **Proposition 1.4.8.** The graph of an upper semicontinuous set-valued map $F\colon X\to Y$ with closed domain and closed values is closed.
> 
> 
> The converse is true if we assume that $Y$ is compact.
> 
> 
> 


Here domain is the set $\operatorname{Dom}(F)=\{x\in X; F(x)\ne\emptyset\}$ (defined on [page 34](https://books.google.com/books?id=cVUl3smcGlcC&pg=PA34)). 


