Cologic of Closed Covers of Compacta and the Pseudo-Arc
Kentarô Yamamoto
TL;DR
This work introduces cologic as a finitary, cover-based logic for compacta, mapping finite quotients of pre-spaces to finite covers by regular closed sets via contact algebras. It develops a Vaught-like theory of types and a cofinal atomicity condition that yields homogeneity results, and then applies these ideas to the pseudo-arc by linking its projective Fraïssé limit to its natural quotient through identical cological theories. The main achievement is showing that the pseudo-arc’s homogeneity can be explained as a logical consequence of cofinal atomicity, thereby justifying the use of pre-spaces and cologic to study compacta. The work also notes limitations, such as the absence of a full compactness theorem for cologic on compacta, while providing a concrete, transferable framework for extending model-theoretic methods to topological continua.
Abstract
A formal system called cologic is proposed for the study of compacta. A counterpart of countable model theory is developed for this system, and it is applied to model theory of the pseudo-arc.