Defective topological spaces: a novel framework for topological spaces with incomplete information

Recent years have seen a surge of interest in representing and reasoning about uncertain information through extended versions of classical mathematical techniques. In this paper, we introduce the concept of defective topological spaces, which is a novel theory motivated by the problem of approaching a topological space with incomplete knowledge of its open sets. In order to address this problem, we define the concepts of exactly and possibly open sets. Furthermore, we investigate the categorical structure of defective topological spaces, and extend several concepts from classical topological spaces, including continuity, convergence, and separation axioms to defective topological spaces.