= Solution
Let $\mathscr H$ be the <set> of <separately continuous> real-valued <functions> on the plane. Take as a <subbasis of a topology>[subbasis] all <sets> $h^{-1}(U)$, where $h\in\mathscr H$ and $U$ is open in the usual <real line>. Unions of finite intersections of these <sets> form a <topology>, and every $h$ is <continuous> for it by construction. Conversely every <topology> making every $h$ <continuous> must contain the <subbasis of a topology>[subbasis], hence this entire generated <topology>. Thus \b[the required coarsest <topology> exists and is unique]: it is the <initial topology> of the family $\mathscr H$.
The family is a <set>, being a <subset> of $\mathbb R^{\mathbb R^2}$, so this construction has no proper-class issue. The <projection map>[coordinate projections] belong to $\mathscr H$; consequently this <topology> contains the ordinary <Euclidean topology>. This last observation will also make <convergence> in the new <topology> imply ordinary coordinatewise <convergence>.
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