Building a propositional logical topological space

To build an observation space, first state the carrier. Our points are complete valuations, not formulas or messages. A is one, zero; B is one, one; C is zero, one; and D is zero, zero under X and Y. Next state the selected, indexed observation formulas. Here the menu contains X and Y. Finally construct a topology from their truth regions. The carrier, the formula presentation and the resulting family of opens are three separate objects. Each chosen formula supplies its true-row region. To generate a topology, include the empty and whole sets and close under unions and finite intersections. The starting regions are generators; the completed collection of opens is the topology. Scan the X column of the semantic table and collect its true rows. They are A and B, so that region is a designated subbasic open. The table describes mathematical truth; displaying it does not simulate delivery of a certificate. Scan the Y column in the same way. Its true rows are B and C. We have now translated the selected columns into two generators, preserving the state labels from the table throughout the translation. If both selected formulas are certified, the valuation lies in their intersection, the singleton containing B. This is a jointly sufficient positive certificate. The intersection narrows the certified event, while the underlying carrier remains unchanged during this construction. If either selected formula is enough, the certified event is their union. A can qualify through X and C through Y. We do not have to claim that one specific test has actually been supplied to explain which events are certifiable. Add the empty and whole sets. The six opens are empty; B alone; A and B; B and C; A, B and C; and the whole carrier. As Lecture 08 showed, further unions and finite intersections add nothing. The observation space is this fixed carrier equipped with this generated topology. The singleton containing A is a perfectly valid classical truth region. It is not open here: every basic positive certificate holding at A also holds at B. A fact can be true without this interface offering a sufficient positive certificate for it. A complete minterm fixes every variable’s truth value, selecting one complete valuation. Joining such descriptions by OR describes their union. Our finite language can define every subset by minterms. If we select formulas for all those subsets, every subset becomes a designated open. The resulting topology is discrete; restricting the observation menu was what made the earlier example informative. More generally, choosing formulas for the open sets of any topology on this finite valuation carrier presents that topology. This does not introduce a newly restricted mathematical class. The value is the common truth-table interpretation of familiar finite constructions. Select just one observation formula: the conjunction X and Y. How many opens are generated on the original four-state carrier? Keep one observation slot: the conjunction is not two separate selected tests. Its one truth region is the singleton containing B. The empty set, that singleton and the whole carrier form the generated topology. A, C and D remain three distinct valuations with the same one-entry false signature. Constructing an observation space means translating selected columns into regions and closing those regions under topology operations. Later we will express the same construction through signatures, order and directed distance without changing the carrier silently.