How a few sets generate a topology

X means dot and Y means stripe. On the original four-card carrier, A has only a dot, B both, C only a stripe, and D neither. A topology contains the empty and whole sets and stays closed under unions and finite intersections. With positive certificates, not receiving a certificate does not establish the opposite statement. Suppose the interface offers two positive certificates: X, true at A and B, and Y, true at B and C. Their truth regions are our starting subsets. We must find the smallest topology containing them on the unchanged four-state carrier. Every topology includes the empty set and the whole carrier. Add those immediately. Our starting family now has four members, but having the endpoints and the generators does not yet guarantee closure under the topology operations. Requiring both certificates means intersecting their regions. Only B satisfies both requirements. The singleton containing B must therefore be open in any topology containing the two starting regions. We have obtained one new open, not a new state. A and C remain in the carrier, although they fail this conjunction. D also remains possible until an actual premise is learned; describing available tests alone removes no point. Allowing either certificate gives the union. A is admitted by X, C by Y, and B by both. This creates the set containing A, B and C. Inclusive OR keeps the overlap rather than discarding it. Together with the endpoints, these are six opens. Further unions and finite intersections produce no new subset. This shows that the list is a topology, rather than merely a collection we happened to write down. Any topology containing the generators must contain their intersection, their union, and the endpoints. We have also checked that these forced sets already form a topology. It follows that no smaller legal topology can contain the two generators. The generating subsets are called a subbasis. Their finite intersections form a basis: the local pieces from which every open is a union. Here those pieces are the whole carrier, A and B, B and C, and B alone. The empty intersection supplies the whole carrier, including when no selected test is required. A complete minterm specifies a truth value for every variable. On our table, X together with not Y is the complete description of A. The minterm X and not Y defines A classically. But not Y is not an available test in this positive menu. Every generated open containing A also contains B, so the singleton containing A is absent from the topology. The singleton minterms from Lecture 05 partition all valuations. They are logical atoms. Calling them a topological basis here would incorrectly make every singleton open and erase the asymmetry our interface is supposed to express. Someone lists only the two generators, the empty set and the whole carrier. Is that already a topology? Name one missing intersection and one missing union before checking the answer. The intersection of the generators is the singleton containing B, which is missing. That one failure already disproves the proposal. A topology must satisfy all the rules, not merely contain some useful events. Their union, the set containing A, B and C, is missing too. Adding both missing sets gives the six-open topology we constructed. A generator list and a topology are different objects, linked by a closure construction.