different notions of continuous maps

Hi,

i just learned the notions of jointly and separately cont. functions.

A function f: X x Y -> A is called

1) jointly cont <=> f is cont. w.r.t. the product topology

2) separ. cont <=> f and are cont.

So i tryed to show 1) => 2).

But all my attempts failed.

Can you help me?

One of my attemts:

Let be cont. then we get an open set ., whereas X' and Y' are open in X, Y respectively.

If we now consider the map and take some U \subset A.

Now we have to proof, is open in Y. But why is this correct in all top. spaces?

Regards

Re: different notions of continuous maps

Yes, it's indeed the idea.

There are some typos in your post. , not and we should have (we know that is open, but the open sets are not always product of open sets). If we denote by , we have and we are done since we know what is.

Re: different notions of continuous maps

Quote:

Originally Posted by

**girdav**

Hello,

I have two questions about your Proof. First, the last equation isn't so obvious for me. Since on the LHS we have a fixed 'x' (depending on the choice of f_x). On the RHS we consider all 'x' (as preimage of f).

Re: different notions of continuous maps

It's wasn't obvious for you because it was false.

To show the result, let and an open set which contains . By definition of the continuity and the product topology, we can find and two open sets with . Now, for we have hence .