Why symmetric distance gives a different topology

A positive certificate establishes that its formula is true. Its absence does not establish falsehood. Symmetric signature distance compares both kinds of answer in the selected profiles. We will compare the opens obtained from these two constructions. Use the explicitly labelled two-point carrier with one positive test X. Its observation topology has three opens: empty, the singleton containing one, and the whole carrier. The singleton containing zero is not positively certifiable. Give the selected coordinate unit weight. The two signatures differ, so their symmetric distance is one. A ball of radius one around zero contains zero alone, because the defining inequality is strict. This ball is open in the metric topology, but it was not open in the positive observation topology. On a finite metric space every singleton has a sufficiently small ball, so the metric topology is discrete. We have not changed the underlying points. We have used two different constructions of opens on the same set. Symmetric disagreement treats the two outcomes alike, while the positive topology retains an asymmetric certification convention. Restore the original four valuations, observing only X. On these original cards, X means dot and Y means stripe. A has only a dot, B both, C only a stripe, and D neither. The fixed bits follow X,Y order, with one meaning true and zero false. Only X is selected for this comparison. A small symmetric ball around A includes B because their distance is zero. It excludes C and D because their signatures differ. With unit weight and strict radius one, this ball is exactly A's signature class. The corresponding ball around C is its class containing C and D. No smaller positive radius separates members of either zero-distance class. Every symmetric ball is a union of whole signature classes, since equal profiles have identical distances to a centre. Conversely sufficiently small balls give each individual class. Unions of those classes are therefore exactly the pseudometric opens. Making both signs of every selected test available also certifies each complete signature pattern. It yields the same partition topology. On this four-state carrier that topology has four opens; it is not discrete because the classes have two points each. The positive one-test space has two quotient points: P represents A and B, while Q represents C and D. Its quotient opens are empty, P alone, and both points. It is T zero but retains one-way separation. Quotienting has removed duplicate profiles, not all order comparisons. The symmetric metric on those same two class labels gives a discrete topology. A metric on the quotient set is therefore not a promise to reproduce the original Kolmogorov quotient topology. Always name which topology accompanies the set. On the original four-state carrier, how many opens come from positive X, from symmetric signature balls, and from adding not X? Which constructions agree, and is the common finer topology discrete? Positive X gives three opens. Symmetric balls and the signed menu both give the four-open partition topology. Neither splits A from B or C from D, so this finer topology is still not discrete on the four valuations. Symmetric distance is useful, but its balls recover two-sided signature information rather than the positive interface in general. To recover the original information direction and its opens, we next measure observations lost from a specified source.