WEBVTT

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An open neighborhood is an open region containing the point.

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

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its support lists the indices answered true.

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Our original X,Y cards are A with a dot only, B

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with both features, C with a stripe only, and D with neither.

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Their first bit answers X and the second Y.

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The paper chooses this direction: v precedes w when every open neighborhood

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of v contains w.

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Read that sentence before reading the arrow.

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Other sources may reverse the specialization convention, so memorizing an arrow without

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its meaning is unsafe.

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We will prove that w preserves every selected positive formula true at

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v.

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It may satisfy additional selected formulas.

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This compares complete semantic profiles; it does not say that a sequence

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of certificates was delivered or that a causal transition occurred.

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Suppose every open containing v contains w.

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Each selected truth region is an open.

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Whenever its formula is true at v, that region contains v and

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therefore w.

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Every positive coordinate of v is consequently positive at w.

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Suppose instead that v's positive support is included in w's.

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Every basic conjunction true at v remains true at w.

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Any union containing v has one such basic witness, which also contains

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w.

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Thus every open containing v contains w.

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Under the menu X, Y, D has empty support, A has the

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first index, C the second, and B both.

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Put D at the bottom, A and C in the middle and

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B at the top.

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Upward means preserving positive information.

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A precedes B because its first positive coordinate survives.

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B does not precede A because the second coordinate would be lost.

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A and C are incomparable, since each lacks one positive coordinate supplied

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by the other.

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In a finite space, intersecting all open neighborhoods of a point still

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gives an open neighborhood.

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There are only finitely many distinct opens.

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This smallest neighborhood consists exactly of points above the chosen point in

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our information comparison.

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A requires X, so its smallest neighborhood contains A and B.

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B requires both tests, so its smallest neighborhood is B alone.

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A point can have a larger smallest neighborhood even though the carrier

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is unchanged.

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C requires Y, giving B and C.

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D requires no positive test.

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The empty conjunction imposes no restriction, so D's smallest neighborhood is

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the whole carrier.

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D is still a point, not the empty set.

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Support inclusion is reflexive and transitive, so this relation on valuations is

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a preorder.

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It can compare two distinct points in both directions when their signatures

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coincide.

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Antisymmetry would require that two-way comparison happens only for the same

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point.

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That requirement fails for distinct valuations with equal signatures.

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Identify points only when both directions hold.

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The classes then carry a partial order, which is also antisymmetric.

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A single comparison in one direction is not a reason to merge

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its endpoints.

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An open is upward closed in this order.

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Conversely an upward-closed subset is a union of the smallest neighborhoods

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of its points, hence open.

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This supplies the finite upper-set representation of the topology.

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Under only X, do A and B precede each other?

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Does D precede A?

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Does A precede D?

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Explain which pair is merged in the quotient.

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A and B have equal supports, so both directions hold and they

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merge.

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D's empty support is included in A's, so D precedes

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A.

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The reverse fails because X is true at A and false at

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D.

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The order, smallest neighborhoods and opens encode the same finite qualitative structure.

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Directed distance will recover this particular direction through zero loss, while its

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positive numerical values will require additional presentation data.