Proof of topology involving Hausdorff spaces

Question

Question: Let X and Y be Hausdorff spaces. Give X X Y the product topology. Then X X Y is a Hausdorff space.

This is what I did, can someone verify this and let me know if I am correct or wrong? Also, kindly let me know if my proof need some changes or modifications due to bad notations.

Any help will be greatly appreciated.

Answer 1

I think you've got it. A couple small things. You reversed $x_i$ and $U_i$ when you chose disjoint $U_i$ containing the $x_i$.

Secondly, the notation for the null set is $\emptyset$, or $\{\}$.

Finally, taking one last peek, you might want to write out the word "and" instead of writing $\land$ in what is basically an English sentence.

I think the substance of your proof is correct.