= Solution
{c}
The <product topology> on $X\times Y$ has as a <basis of a topology> the rectangles $U\times V$ with $U$ open in $X$ and $V$ open in $Y$; arbitrary unions of such rectangles are its open sets.
For distinct points $(x_1,y_1),(x_2,y_2)$, at least one coordinate differs. If $x_1\ne x_2$, disjoint open neighbourhoods $U_1,U_2$ in the <Hausdorff space> $X$ give disjoint open neighbourhoods $U_1\times Y,U_2\times Y$ of the two points. If the second coordinates differ instead, use $X\times V_1,X\times V_2$. Thus \b[a product of two Hausdorff spaces is Hausdorff].
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