Limit of sequence of sets - Some paradoxical facts

Solution 1:

The definition of liminf used by Rockafellar and Wets is different from the definition you quote at the beginning of your post (which does not require any structure on the reference space, topological or anything else) hence the results are different.

The notion of liminf defined at the beginning of your post, which I will call set theoretical and denote $\mathrm{set}\!\!-\!\!\liminf$, does not require any structure on the reference space, and in particular no topology is involved. R&W's notion of liminf, which I will call topological and denote $\mathrm{top}\!\!-\!\!\liminf$, does require a topology since, according to a comment of yours, it involves convergent sequences (in your comment, $x_v\to x$) and to define convergence one needs a topology. Hence Propositions 1 and 3A on the one hand, and Propositions 2 and 3B on the other hand, are based on two different definitions of liminf. For every $(C_n)$, $$ \mathrm{set}\!\!-\!\!\liminf C_n\subseteq\mathrm{top}\!\!-\!\!\liminf C_n, $$ and these can be quite different. For example, consider the real line $\mathbb R$ equipped with the usual topology, and $C_{2n}=D$ and $C_{2n+1}=\mathbb R\setminus D$ for every $n$, where $D$ is dense and co-dense, for example $D=\mathbb Q$. Then $\mathrm{set}\!\!-\!\!\liminf C_n$ is the empty set and $\mathrm{top}\!\!-\!\!\liminf C_n$ is $\mathbb R$.