Continuous functions have closed graphs

[quasar987]

Homework Statement

How is the theorem "Continuous functions have closed graphs" proven in the setting of a general topological space? (assuming the theorem is still valid?)

 

[quasar987]

Actually I found a 6 months old thread in which the OP asks help to prove that a continuous map from a topological space X to a Hausdorff space Y has a closed graph

(https://www.physicsforums.com/showthread.php?t=171861&highlight=closed+graph)

I haven't been able to prove this version either though. I suppose it deals with sequences and the Hausdorff property is there to ensure the uniqueness of a limit, but I'm not familiar enough with sequence theory in general topological spaces to construct an argument.

 

[HallsofIvy]

You might want to look at
http://en.wikipedia.org/wiki/Closed_graph_theorem
http://mathworld.wolfram.com/ClosedGraphTheorem.html

 

[quasar987]

I am interested in the converse to the closed graph theorem.

The closed graph theorem says "E and F Banach, and T linear with a closed graph. Then T is continuous."

Very restrictive. Under which conditions is the converse true? I have shown that for a map T btw metric spaces, "T continuous ==> T has a closed graph". But I'm sure it holds under weaker hypotheses!

 

[morphism]

I just scribbled some stuff down, and I think I managed to prove that if Y isn't Hausdorff, then there exists a topological space X and a continuous map f:X->Y whose graph isn't closed. So, provided what I did was alright (and I think it is - it wasn't a very intricate argument, but based on nets), this means that saying "the graph of any continuous function from X to Y is closed" is equivalent to saying "Y is Hausdorff".

I don't think this is very surprising. There are a few other results of a similar flavor, e.g. a space is Hausdorff iff its diagonal is closed, a Hausdorff space is metrizable iff its diagonal is a zero set, etc.