 Solution

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.