WEBVTT

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A positive certificate establishes that its formula is true.

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Its absence does not establish falsehood.

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Symmetric signature distance compares both kinds of answer in the selected profiles.

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We will compare the opens obtained from these two constructions.

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Use the explicitly labelled two-point carrier with one positive test X.

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Its observation topology has three opens: empty, the singleton containing one, and

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the whole carrier.

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The singleton containing zero is not positively certifiable.

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Give the selected coordinate unit weight.

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The two signatures differ, so their symmetric distance is one.

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A ball of radius one around zero contains zero alone, because the

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defining inequality is strict.

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This ball is open in the metric topology, but it was not

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open in the positive observation topology.

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On a finite metric space every singleton has a sufficiently small ball,

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so the metric topology is discrete.

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We have not changed the underlying points.

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We have used two different constructions of opens on the same set.

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Symmetric disagreement treats the two outcomes alike, while the positive topology retains

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an asymmetric certification convention.

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Restore the original four valuations, observing only X.

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On these original cards, X means dot and Y means stripe.

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A has only a dot, B both, C only a stripe, and

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D neither.

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The fixed bits follow X,Y order, with one meaning true and

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zero false.

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Only X is selected for this comparison.

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A small symmetric ball around A includes B because their distance is

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zero.

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It excludes C and D because their signatures differ.

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With unit weight and strict radius one, this ball is exactly A'

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s signature class.

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The corresponding ball around C is its class containing C and D.

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No smaller positive radius separates members of either zero-distance class.

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Every symmetric ball is a union of whole signature classes, since equal

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profiles have identical distances to a centre.

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Conversely sufficiently small balls give each individual class.

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Unions of those classes are therefore exactly the pseudometric opens.

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Making both signs of every selected test available also certifies each complete

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signature pattern.

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It yields the same partition topology.

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On this four-state carrier that topology has four opens; it is

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not discrete because the classes have two points each.

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The positive one-test space has two quotient points: P represents A

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and B, while Q represents C and D.

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Its quotient opens are empty, P alone, and both points.

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It is T zero but retains one-way separation.

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Quotienting has removed duplicate profiles, not all order comparisons.

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The symmetric metric on those same two class labels gives a discrete

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topology.

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A metric on the quotient set is therefore not a promise to

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reproduce the original Kolmogorov quotient topology.

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Always name which topology accompanies the set.

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On the original four-state carrier, how many opens come from positive

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X, from symmetric signature balls, and from adding not X?

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Which constructions agree, and is the common finer topology discrete?

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Positive X gives three opens.

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Symmetric balls and the signed menu both give the four-open partition

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topology.

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Neither splits A from B or C from D, so this finer

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topology is still not discrete on the four valuations.

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Symmetric distance is useful, but its balls recover two-sided signature information

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rather than the positive interface in general.

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To recover the original information direction and its opens, we next measure

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observations lost from a specified source.