A topological characterization of weakness
We are interested in the relation between weak and strong temporal operators. We would like to find a characterization that shows what it means for an operator to be the weak or strong version of another operator, or more generally for a formula to be a weak or strong version of another formula. We show that the weak version of a formula is not the same as Alpern and Schneider's safety component. By working over an extended alphabet, we show that their topological characterization of safety can be adapted to obtain a topological characterization of weakness. We study the resulting topology and the relations between weak and strong formulas. Finally, we apply the method to show the internal consistency of a logic containing both weak and strong versions of regular expressions.