How directed distance recovers the topology

Directed loss adds the weights of selected true-to-false changes from source to destination. Gains contribute nothing, and the source is the first argument of q. On our original X,Y table, A is one-zero, B one-one, C zero-one and D zero-zero. One means true and zero means false. A point’s smallest open neighborhood contains exactly the worlds that preserve its selected positives. A forward ball centred at v contains destinations reached with directed loss strictly less than the radius. Keep v in the first argument. Reversing those arguments generally makes a different set and does not prove the topology statement we want. Use observations X and Y with weights one and three, respectively. Any positive loss then costs at least one. Choose a positive radius no greater than one. Because the inequality is strict, a loss costing exactly one is excluded from the ball. A has X true and Y false. Moving to B loses no true coordinate, so B belongs to every positive-radius forward ball around A. Moving to C or D loses X and costs one. At radius one, the cost-one destinations are excluded. The ball is A and B, exactly the smallest positive open neighborhood of A. It is not the singleton symmetric ball from the preceding comparison. B has both observations true. Any other valuation loses at least one positive coordinate, so a small forward ball at B contains only B. This agrees with the conjunction certificate for both observations. C retains its Y observation at B and C. D has no positive observations to lose, so every destination has zero loss from D. Its small forward ball is the whole carrier, exactly as its smallest neighborhood requires. Suppose w lies in a forward ball and u preserves all positive observations at w. Any source observation retained at w is retained at u. Thus u cannot lose more of the centre's observations than w loses. Consequently u also lies in the ball. Every forward ball is upward closed in the information preorder, hence open in the finite observation topology. This argument covers every radius, not only the small one we used for computation. For our finite, nonempty list of strictly positive weights, choose a positive radius no greater than the smallest weight. At such a radius only zero losses are admitted. Zero loss is support inclusion, so the ball is exactly the smallest neighborhood. Every open is a union of these neighborhoods, which proves the reverse containment between the two topologies. If there are no selected observations, there is no minimum weight to take. Handle this case separately: all directed distances are zero, every positive-radius ball is the whole carrier, and the generated topology is indiscrete. The result still holds. Which destinations lie in the forward ball around A at radius one? What changes at radius one point one? Use the strict inequality, and keep the source fixed. A and B have zero loss from A, while C and D cost one. At radius one only A and B enter. At radius one point one, all four enter because the cost-one destinations now satisfy the strict inequality. Forward balls recover the positive observation topology because their small forms are precisely its smallest neighborhoods. The topology fixes those qualitative neighborhoods, but it still does not determine the positive distances or the chosen weights.