WEBVTT

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This extension asks whether a finite observation space splits into disconnected pieces.

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The connectedness discussion was present in the earlier manuscript and is not

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a standalone claim of the v0.

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9 core.

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We retain it as labelled enrichment using the specialization convention already established.

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Use the original X,Y assignments: A is one-zero, B one-

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

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A selected signature gives the truth answers to the chosen formula slots;

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its support lists the true slots.

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One point precedes another when all its selected positives are retained.

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Different original points may have the same signature.

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One means true and zero means false.

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A space is disconnected if it is the union of two disjoint

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nonempty open subsets.

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Each side is then also closed because its complement is the other

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open side.

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A subset both open and closed is called clopen.

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Use each original point as a vertex.

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Join two distinct points whenever either precedes the other in the specialization

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preorder.

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Make the edge undirected for this connectedness test, even though the underlying

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information comparison may be one-way.

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Distinct points with equal signatures precede each other and must also be

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joined.

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Omitting these edges would falsely split an indiscrete block into isolated graph

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vertices.

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We have not assumed that the original space is T zero.

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The positive X,Y cover diagram links D to A and C,

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and those to B.

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The full comparability graph also joins D directly to B, but adding

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that edge does not change the connected components.

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Following undirected edges connects every point.

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Direct comparability between A and C is unnecessary: a path through another

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point is enough.

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The graph has one component, and the topology is connected.

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A lack of an open separating A in one direction from B

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does not mean the whole space is disconnected.

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Connectedness and pairwise distinguishability answer different questions.

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Keep the four valuations and make both signs of X available.

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A and B form one equal-signature block, and C and D

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another.

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Each pair is joined internally; there is no comparison edge crossing between

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the two blocks.

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The two blocks are clopen and are the graph components.

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Each block is connected internally despite containing two indistinguishable points.

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Confusing connected components with singleton distinguishability would lose exactly this example.

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If a point lies in one graph component, every comparable point is

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in that component.

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The component is therefore both an upper and a lower set.

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In the finite upper-set topology this makes it clopen.

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If two clopen sides had comparable points across them, upward closure of

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one side or of its complement would force a point across the

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separation.

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No comparison can cross.

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A graph component therefore cannot be divided by a clopen separation.

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For a new check, use three tests: X, Y, and the negation

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of the whole conjunction X and Y.

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Can B connect by a comparison edge to A, C or D

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under this menu?

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Use the displayed signatures and predict the connected components before the answer.

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B has signature one, one, zero.

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The others have the third coordinate true, so neither inclusion direction connects

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B to them.

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D precedes A and C, connecting those three.

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The components are B alone and A, C and D together.

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The undirected comparison graph detects connected components of these finite spaces, including

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equivalent distinct points.

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It does not measure metric distance, and its components must not be

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identified with the separate test-compatibility complex introduced next.