This course is an introduction to topology and an exploration of some of its applications in modal logic, with particular focus on epistemic logics. Some basic background in modal logic will be helpful, but is not essential; no background in topology is assumed.
We begin by motivating the notion of a topological space using a variety of metaphors and intuitions. Then, connecting the formal definition of a set's interior to the abstract concept of "robustness", we introduce topological semantics for the basic modal language, investigate its relationship with the more standard relational semantics, and establish the foundational result that S4 is "the logic of space" (i.e., sound and complete with respect to the class of all topological spaces). Rather than using logic to study space, however, we focus on using space to study logic—specifically, epistemic logic.
From this perspective, roughly speaking, the spatial notion of "nearness" is co-opted as a means of representing uncertainty. Viewing epistemology through the lens of topology highlights the distinction between the known and the knowable, between fact and measurement. To more fully incorporate this conceptual framework into our analysis, we introduce topological subset space semantics, which allows us to manipulate separately the state of the world and the epistemic state of the agent. We close with a look at some recent applications of these models and areas of active research, such as the interpretation of public announcements, and the extension to multi-agent settings.