Is the Graph of a Continuous Map from X to Y a Closed Subset of XxY?

[pivoxa15]

Homework Statement

Let f:X->Y be a cts map from a topological space X to a Hausdorff space Y. Prove that the graph L={(x,y) in XxY: y=f(x)} is a closed subset of XxY.

The Attempt at a Solution

Hausdorff space are linked with open sets so how do you prove closeness in XxY?

 

[matt grime]

This is not important but since open sets are linked to closed sets, then anything to do with closed sets has some relation with open sets. You understand that a set is open if and only if its complement is closed?

So, start with L. What does it mean to be closed? There are many things to try, so write a few of them down and think about it for a while (perhaps a day or so, if need be - answers don't just magically appear instantly in people's minds).

 

[pivoxa15]

There are many things to try, so write a few of them down and think about it for a while (perhaps a day or so, if need be - answers don't just magically appear instantly in people's minds).

If only the assignment is not due tomorrow morning.

 

[matt grime]

L is closed if and only if its complement is open. Suppose that the complement is not open. What does that mean?

 

[complexnumber]

L is closed if and only if its complement is open. Suppose that the complement is not open. What does that mean?

Can you give more hint?

Why can't we use the proof in Banach space in the following link?
http://myyn.org/m/article/proof-of-closed-graph-theorem/