I need help with the following proof. Would be thankful for any help.
If $X, Y$ are nonempty topological spaces. Given the $X \times Y$ is Hausdorff, show that $X$ and $Y$ each are Hausdorff spaces.
I tried to look at given points $x_1, x_2$ in $X$ with $x_1 \ne x_2$ and $y_1, y_2$ in $Y$ with $y_1 \ne y_2$. Then I used the definition of the product topology and the Hausdorff property of $X \times Y$, to find open neighborhoods $U_1 \times V_1$ of $(x_1, y_1)$ and $U_2 \times V_2$ of $(x_2, y_2)$, $U_1, U_2$ open in $X$ and $V_1, V_2$ open in $Y$. Then I used the Hausdorff property to say that the intersection of $U_1 \times V_1$ and $U_2 \times V_2$ is empty. This also means $(U_1 \cap U_2) \times (V_1 \cap V_2)$ is empty. This means either $U_1 \cap U_2$ or $V_1 \cap V_2$ is empty. I can't really figure out how to prove that if $U_1 \cap U_2$ is empty and $V_1 \cap V_2$ is not empty, that $Y$ is Hausdorff.