I am studying set-valued analysis, and I saw three version of this Closed Graph Theorem.
Version 1: (what I was taught in class)
Let $\Gamma: X \to Y$ be a correspondence. If $\Gamma$ is closed-valued and upper hemi-continuous, then $Gr(\Gamma)$ is closed.
Version 2: (from Set-Valued Analysis by Jean-Pierre Aubin, Hélène Frankowska)
Let $\Gamma: X \to Y$ be an upper hemi-continuous correspondence with closed domain and is closed-valued, then $Gr(\Gamma)$ is closed.
Version 3: (from Infinite Dimensional Analysis: A Hitchhiker's Guide by Charlambos Aliprantis and Kim C. Border)
A correspondence with compact Hausdorff range space is closed if and only if it is upper hemi-continuous and closed-valued.
I want to know if they are equivalent to each other. I'm not sure how to prove it or give a counterexample. Could someone please help me with it? Thanks so much in advance!
Here is a reference about related definitions and results that might be helpful: Prove the (path-) connectedness of the graph of a compact- and convex-valued upper hemi-continuous correspondence.