Suppose I have a topological space whose underlying set $X$ has a partial ordering on it. A prior this partial ordering has no relation to the topology.
Now, take any meet semi-lattice $I \subseteq X,$ and an arbitrary set of open sets $\{A_i\}_{i\in I}$ satisfying $i\in A_i$ for all $i\in I.$
What are the conditions (on the topology or the partial ordering) that guarantee the existence of another family of open sets $\{B_i\}_{i\in I}$ satisfying $i\in B_i\subseteq A_i$ and $$B_i\cap B_j\subseteq B_{i\wedge j}$$ for all $i\in I$ ?