WEBVTT

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Directed loss adds the weights of selected true-to-false changes from

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source to destination.

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Gains contribute nothing, and the source is the first argument of q.

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On our original X,Y table, A is one-zero, B one-

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one, C zero-one and D zero-zero.

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One means true and zero means false.

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A point’s smallest open neighborhood contains exactly the worlds that preserve

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its selected positives.

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A forward ball centred at v contains destinations reached with directed loss

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strictly less than the radius.

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Keep v in the first argument.

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Reversing those arguments generally makes a different set and does not prove

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the topology statement we want.

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Use observations X and Y with weights one and three, respectively.

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Any positive loss then costs at least one.

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Choose a positive radius no greater than one.

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Because the inequality is strict, a loss costing exactly one is excluded

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from the ball.

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A has X true and Y false.

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Moving to B loses no true coordinate, so B belongs to every

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positive-radius forward ball around A.

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Moving to C or D loses X and costs one.

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At radius one, the cost-one destinations are excluded.

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The ball is A and B, exactly the smallest positive open neighborhood

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of A.

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It is not the singleton symmetric ball from the preceding comparison.

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B has both observations true.

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Any other valuation loses at least one positive coordinate, so a small

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forward ball at B contains only B.

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This agrees with the conjunction certificate for both observations.

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C retains its Y observation at B and C.

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D has no positive observations to lose, so every destination has zero

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loss from D.

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Its small forward ball is the whole carrier, exactly as its smallest

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neighborhood requires.

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Suppose w lies in a forward ball and u preserves all positive

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observations at w.

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Any source observation retained at w is retained at u.

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Thus u cannot lose more of the centre's observations than w

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loses.

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Consequently u also lies in the ball.

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Every forward ball is upward closed in the information preorder, hence open

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in the finite observation topology.

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This argument covers every radius, not only the small one we used

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for computation.

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For our finite, nonempty list of strictly positive weights, choose a positive

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radius no greater than the smallest weight.

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At such a radius only zero losses are admitted.

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Zero loss is support inclusion, so the ball is exactly the smallest

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neighborhood.

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Every open is a union of these neighborhoods, which proves the reverse

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containment between the two topologies.

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If there are no selected observations, there is no minimum weight to

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take.

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Handle this case separately: all directed distances are zero, every positive-radius

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ball is the whole carrier, and the generated topology is indiscrete.

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The result still holds.

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Which destinations lie in the forward ball around A at radius one?

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What changes at radius one point one?

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Use the strict inequality, and keep the source fixed.

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A and B have zero loss from A, while C and D

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cost one.

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At radius one only A and B enter.

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At radius one point one, all four enter because the cost-one

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destinations now satisfy the strict inequality.

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Forward balls recover the positive observation topology because their small forms are

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precisely its smallest neighborhoods.

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The topology fixes those qualitative neighborhoods, but it still does not determine

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the positive distances or the chosen weights.