Me again.(adsbygoogle = window.adsbygoogle || []).push({});

Problem.Let X be a topological space, and Y a T2-space (i.e. a Haussdorf topological space). Let f : X --> Y be a continuous function. One needs to show that the graph of , i.e. the set G = {(x, f(x)) : x is in X} is closed in X x Y.

Attempt of proof. To show what we need to show, we have to prove that the complement of G is open. Now, since Y is a Haussdorf space, one can show that it is a T1-space too, i.e. that every set containing a single point is closed. So, for every x, {f(x)} is closed in Y. Further on, since f is continuous, the preimage of every closed set is a closed set too, so, for any f(x), {x} is closed in X. I feel I'm getting very warm, but I got stuck anyway. The complement of G in X x Y is X x Y \ G = X x Y \ U (x, f(x)) (where the union goes through all x in X) = [itex]\cap[/itex] [(X x Y) \ (x, f(x))].Now,if X and Y are T1-spaces, can one show that X x Y is a T1 space, too? But then we'd in general have an infinite intersection of open sets, which doesn't need to be open.

Thanks in advance for a push.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Showing a graph is closed

**Physics Forums | Science Articles, Homework Help, Discussion**