Rectangularity of every product-open set

Every open subset of a product of two topological spaces is a rectangle exactly when at least one factor has the indiscrete topology. For necessity, if both factors have nonempty proper open sets $U⊊X$, $V⊊Y$, then $(U\times Y)\cup (X\times V)$ is open, projects onto both full factors, and omits every point of $(X\setminus U)\times (Y\setminus V)$. A rectangle with both full projections would be the whole product, a contradiction. For sufficiency, if $X$ is indiscrete, every basic open rectangle is $X\times V$ or empty; an arbitrary union is $X\times \bigcup V$. The same argument works when $Y$ is indiscrete. Empty factors cause no exception because their topology is indiscrete.