WEBVTT

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To build an observation space, first state the carrier.

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Our points are complete valuations, not formulas or messages.

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A is one, zero; B is one, one; C is zero, one;

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and D is zero, zero under X and Y.

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Next state the selected, indexed observation formulas.

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Here the menu contains X and Y.

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Finally construct a topology from their truth regions.

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The carrier, the formula presentation and the resulting family of opens are

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three separate objects.

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Each chosen formula supplies its true-row region.

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To generate a topology, include the empty and whole sets and close

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under unions and finite intersections.

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The starting regions are generators; the completed collection of opens is the

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topology.

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Scan the X column of the semantic table and collect its true

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rows.

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They are A and B, so that region is a designated subbasic

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open.

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The table describes mathematical truth; displaying it does not simulate delivery of

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a certificate.

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Scan the Y column in the same way.

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Its true rows are B and C.

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We have now translated the selected columns into two generators, preserving the

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state labels from the table throughout the translation.

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If both selected formulas are certified, the valuation lies in their intersection,

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the singleton containing B.

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This is a jointly sufficient positive certificate.

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The intersection narrows the certified event, while the underlying carrier remains unchanged

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during this construction.

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If either selected formula is enough, the certified event is their union.

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A can qualify through X and C through Y.

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We do not have to claim that one specific test has actually

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been supplied to explain which events are certifiable.

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Add the empty and whole sets.

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The six opens are empty; B alone; A and B; B and

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C; A, B and C; and the whole carrier.

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As Lecture 08 showed, further unions and finite intersections add nothing.

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The observation space is this fixed carrier equipped with this generated topology.

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The singleton containing A is a perfectly valid classical truth region.

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It is not open here: every basic positive certificate holding at A

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also holds at B.

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A fact can be true without this interface offering a sufficient positive

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certificate for it.

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A complete minterm fixes every variable’s truth value, selecting one complete

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valuation.

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Joining such descriptions by OR describes their union.

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Our finite language can define every subset by minterms.

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If we select formulas for all those subsets, every subset becomes a

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designated open.

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The resulting topology is discrete; restricting the observation menu was what made

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the earlier example informative.

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More generally, choosing formulas for the open sets of any topology on

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this finite valuation carrier presents that topology.

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This does not introduce a newly restricted mathematical class.

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The value is the common truth-table interpretation of familiar finite constructions.

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Select just one observation formula: the conjunction X and Y.

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How many opens are generated on the original four-state carrier?

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Keep one observation slot: the conjunction is not two separate selected tests.

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Its one truth region is the singleton containing B.

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The empty set, that singleton and the whole carrier form the generated

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topology.

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A, C and D remain three distinct valuations with the same one-

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entry false signature.

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Constructing an observation space means translating selected columns into regions and closing

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those regions under topology operations.

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Later we will express the same construction through signatures, order and directed

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distance without changing the carrier silently.