Is there a name for this lattice theoretic property? Suppose that $\langle L, \wedge, 0 \rangle$ is a lower semilattice with least element $0$. For preliminary notation, for all $b \in L$ define ${\downarrow}b = \{ a \in L \colon a \leq b \} $. Here $\leq$ is the partial order induced by the meet operation. The property I am interested in I call swathing. For all $s \in L$ and all $A \subseteq {\downarrow}s$ we will say that $s$ swathes $A$ if and only if: for all $b \in {\downarrow}s$ if, for all $a \in A$ we have $a \wedge b = 0$ then we also have $b = 0$.
My interest in this property stems from the following observation concerning convex sets in ${\mathbb{R}}^{n}$. The meet operation will be intersection. A convex set $O$ is open in the usual topology if and only if for all convex $C \subseteq {\mathbb{R}}^{n}$ if neither $C \cap O$ nor $C \smallsetminus O$ is empty then there exists a nonempty convex set $S \subseteq C \smallsetminus O$ that satisfies $O \cup S$ swathes $\{ O, S \} $.
Any use of the real numbers is hidden in the construction of the convex sets. Once you prove that if two convex sets $O_{0}$ and $O_{1}$ have this property then so does $O_{0} \cap O_{1}$. This property gives us a basis for the usual topology.
With a little additional fussiness we can work in a semilattice as described above and construct what amounts to a basis for a locale.
Is there an commonly used name for this property.

In On the duality of cardinal invariants, Top. Proc. 2, no. 2 (1977) pp. 563-582 I used the ad hoc terminology $A F\text{-}covers\;s$, but that was chosen to suit a very specific context. Outside that context my inclination would be to say that $A$ is dense in (or below) $s$. I don’t know any standard term, but I’ve not worked much in this area, and there could certainly be one of which I’m not aware.
