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Put $X={\mathbb{R}}_{+}^L$ and identify a continuous preference relation $R$ with its closed graph $G_R=\{(x,y)\in X\times X:xRy\}$. The closed convergence of preference relations is the topology induced by the Fell topology on these closed subsets of $X\times X$.

Concretely, a subbasis consists of the setsFinite intersections of these requirements form a basis, restricted to the specified complete, transitive, continuous preference relations. Thus a neighborhood may require some comparisons in prescribed open regions and forbid any comparison in prescribed compact regions. The use of compact sets in the avoidance condition is what makes this the Fell topology, rather than a stronger hyperspace topology on an unbounded consumption space.

There are two equivalent graph-limit requirements. Both are necessary. The upper condition is that, for every subsequence $n_j$ and convergent pairs $x_j\to x$, $y_j\to y$,The lower condition is that every limiting comparison can be approximated: for every $x⪰y$, there exist $x_n\to x$ and $y_n\to y$ withThe upper condition excludes spurious limiting comparisons; the lower condition prevents genuine limiting comparisons from disappearing. The subsequence quantifier in the upper condition is important: a violation need occur along only some of the indices. On the locally compact separable metric space $X\times X$, these two conditions are exactly closed convergence of preference relations.