Suppose that $(A,\le)$ is a complete lattice, that means $(A,\wedge,\vee)$ is a lattice which satisfies $\forall B \subseteq A[\bigwedge B\text{ and }\bigvee B\text{ exist}].$ And of course $(\wp(A),\subseteq)$, in which $\wp(A)$ is the powerset of $A$, is a complete lattice too (let $\bigcap \emptyset=A$). Furthermore, Let $(D,\sqsubseteq)$ be a directed set, and $P \colon D \to \wp(A)$ s.t. $\forall \alpha,\beta \in D[\alpha \le \beta \Rightarrow P_{\alpha} \supseteq P_{\beta}]$. Then if $\bigcap_{\alpha \in D}P_{\alpha} \ne \emptyset$, do
$\bigvee \bigcap_{\alpha \in D}P_{\alpha}=\bigwedge_{\alpha \in D}\bigvee P_{\alpha}$?
$\bigwedge \bigcap_{\alpha \in D}P_{\alpha}=\bigvee_{\alpha \in D}\bigwedge P_{\alpha}$?
That is, in discrete topology, are the limit superior and limit inferior of a directed net exactly the supremum and infimum of this net's limit set respectively?