Why the pullback has the induced topology

R and S are premise regions inside the same ambient world space. For their inclusion maps, a matching pair has the same world in both coordinates. Such pairs are exactly v,v for worlds in the intersection, which we call T. Equality here is equality of valuations, even when different valuations have the same observed signature. The matching pairs from the previous lecture correspond to the intersection points. To identify the two objects as topological spaces, we must also compare their opens. The carrier alone does not determine a topology. A bijection pairs the points on the two sides one-to-one, leaving none out. An inverse image consists of source points landing in a chosen target event. Continuity requires every open target event to have an open inverse image. A homeomorphism is a bijection that is continuous in both directions. It preserves the topological structure, not merely the number of points. We will use the diagonal map from the intersection to its compatible pairs. A subspace R inherits regions R intersect U from ambient opens U. In a product, an open rectangle requires an open condition on each coordinate; unions of these rectangles give the product opens. A coordinate projection simply reads one component of the pair. Each truth region inherits its topology by intersection with ambient opens. The intersection region inherits the same way. On the matching-pair set, use the subspace topology from the product of the two premise subspaces. The diagonal map has two continuous inclusion coordinates, so it is continuous into the product and into the matching-pair subspace. Either coordinate projection restricted to the matching pairs is its continuous inverse. Thus the identification is a homeomorphism. For the separate illustration, use the original four X,Y values: A is one-zero, B one-one, C zero-one and D zero-zero. Select positive X and Y. We will call the illustrative subspace J, keeping T for the general intersection R intersect S. One means true and zero means false. For a separate subspace example, restrict the positive four-state topology to A and B, and call this subspace J. The ambient Y region meets J at B, so B alone must be open in the induced topology. The induced opens are empty, B and A and B. If we instead assign only empty and the whole subspace as opens, the inclusion into the ambient space is no longer continuous. The ambient open Y region pulls back to B, which that arbitrary replacement topology does not contain. Now return to the general intersection T, equal to R intersect S. The separate J example is finished; R and S are again arbitrary premise subspaces in the same ambient space. Suppose two continuous maps from another space agree after inclusion into the ambient carrier. At every input z their output points are the same valuation. That valuation belongs to both premise regions, hence to their intersection. Assign this common value as the map into the intersection. It is the only possible assignment that can reproduce both given maps. Projecting the corresponding matching pair gives back f and g, so the required diagram factors. An open of the intersection is an ambient open restricted to T. Its inverse image under the common-value map is the inverse image under f of the corresponding open in R. That set is open because f is continuous. Any other mediator reproducing the same included values must assign the same common valuation at every input. This proves uniqueness as well as existence and continuity. These three requirements are the universal property in this inclusion case. For this check, take J to be the subspace containing A and B. The ambient Y region contains B and C and is open. What is its inverse image under the inclusion of J into the ambient space? Why would giving J only the empty and whole opens make that inclusion fail continuity? The required inverse image is B. It is open in the induced subspace topology but absent from the indiscrete replacement. The inclusion would fail continuity, so that replacement cannot simply stand in for the topological pullback. The pullback of truth-subspace inclusions is their intersection with its induced topology. The universal property records a unique continuous common-value map. It packages ordinary conjunction structurally; it does not alter classical validity or license arbitrary topologies on truth regions.