# A weaker locally convex topology on a pospace

A pospace is a topological space $X$ endowed with a closed partial order $\le$. A pospace $X$ is locally convex if it has a base of the topology consisting of open order-convex sets. A subset $A$ of a pospace $X$ is called order-convex if for any points $a,b\in A$ the order interval $[a,b]=\{x\in X:a\le x\le b\}$ is contained in $X$. A subset $A$ of a pospace $X$ is called upper (resp. lower) if for every $a\in A$ the set ${\uparrow}a=\{x\in X:a\le x\}$ (resp. ${\downarrow}a=\{x\in X:x\le a\}$) is contained in $A$. It is clear that each upper or lower set is order-convex.

Each pospace $(X,\tau)$ carries a weaker locally convex topology $weak_\diamondsuit$ generated by a base consiting of open order-convex sets in $X$. Another weaker locally convex topology is the topology $weak_{\uparrow\downarrow}$, generated by the subbase consisting of open sets which are upper or lower. The topology $weak_{\uparrow\downarrow}$ contains the interval topology $weak_{\leftrightarrow}$ on $X$, generated by the subbase consisting of the sets $X\setminus{\uparrow}x$ and $X\setminus{\downarrow}x$ where $x\in X$. It is clear that $$weak_{\leftrightarrow}\subset weak_{\uparrow\downarrow}\subset weak_{\diamondsuit}\subset\tau.$$

The interval topology has been mentioned in the Theory of continuous lattices and domains. I am interested if anybody considered the weak locally convex topologies $weak_{\uparrow\downarrow}$ and $weak_{\diamondsuit}$. Maybe they have some special names (or some standard notations)? If yes, could you provide me with suitable references.