Editing Closed graph theorem (section)

== Relation to the open mapping theorem ==
Often, the closed graph theorems are obtained as corollaries of the [[open mapping theorem]]s in the following way.<ref name="Tao" /><ref>{{cite arXiv | eprint=2403.03904 | last1=Noll | first1=Dominikus | title=Topological spaces satisfying a closed graph theorem | date=2024 | class=math.GN }}</ref> Let <math>f : X \to Y</math> be any map. Then it factors as
:<math>f: X \overset{i}\to \Gamma_f \overset{q}\to Y</math>.
Now, <math>i</math> is the inverse of the projection <math>p: \Gamma_f \to X</math>. So, if the open mapping theorem holds for <math>p</math>; i.e., <math>p</math> is an open mapping, then <math>i</math> is continuous and then <math>f</math> is continuous (as the composition of continuous maps).

For example, the above argument applies if <math>f</math> is a linear operator between Banach spaces with closed graph, or if <math>f</math> is a map with closed graph between compact Hausdorff spaces.