= 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> $U\subsetneq X$, $V\subsetneq Y$, 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.
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