Topology: two product space questions

[Damascus Road]

Greetings all,
doing more problems for my test tomorrow. I'm not sure how to start these two..

1.) I'm trying to show that if X and Y are Hausdorff spaces, then the product space X x Y is also Hausdorff.

So, I know that I must have distinct x1 and x2 $$\in$$ X, with disjoint neighborhoods U1 and U2 so that x1 $$\in$$ U1 and x2 $$\in$$ U2.
Similar with y1 and y2 in Y with neighborhoods W1 and W2.

Then what? The product topology seems to be mostly defined by the basis of the product or the basis' of the spaces... but I'm not sure how to draw my next step from that.

2.) This probably requires the same insight as the first... I'm trying to show that if A is closed in X and B is closed in Y, then A x B is closed in X x Y.

Thanks!

 

[Dick]

If you have two points in X x Y you can write them as (x1,y1) and (x2,y2) where the x's are in X and they y's are in Y. If they are distinct, then either x1 is not equal to x2 or y1 is not equal to y2.