## The closed graph theorem in various categories

Given a function $latex {f: X \rightarrow Y}&fg=000000$ between two sets $latex {X, Y}&fg=000000$, we can form the graph

$latex \displaystyle \Sigma := \{ (x,f(x)): x\in X \},&fg=000000$

which is a subset of the Cartesian product $latex {X \times Y}&fg=000000$.

There are a number of closed graph theorems" in mathematics which relate the regularity properties of the function $latex {f}&fg=000000$ with the closure properties of the graph $latex {\Sigma}&fg=000000$, assuming some completeness" properties of the domain $latex {X}&fg=000000$ and range $latex {Y}&fg=000000$.