Revision with unchanged content. In this highly original text, Christopher Steinsvold explores an alternative semantics for logics of rational belief. Topologies, as mathematical objects, are typically interpreted in terms of space; here topologies are re-interpreted in terms of an agent with rational beliefs. The topological semantics tells us that the agent can never, in principle, know everything; that the agent's beliefs can never be complete. A number of elegant completeness proofs are given for a variety of logics of rational belief. Beyond this, the author explores the philosophical question of why our beliefs can never be complete, and considers the possibility that a totality of truths is a dialethia. This work will be of interest to all philosophers interested in epistemology, and modal logicians as well.