  In recent work, Stalnaker proposes a logical framework in which belief is
realized as a weakened form of knowledge. Building on Stalnaker's core
insights, and using frameworks developed in previous work by Bjorndahl and
Baltag et al., we employ topological tools to refine and, we argue, improve on
this analysis. The structure of topological subset spaces allows for a natural
distinction between what is known and (roughly speaking) what is knowable; we
argue that the foundational axioms of Stalnaker's system rely intuitively on
both of these notions. More precisely, we argue that the plausibility of the
principles Stalnaker proposes relating knowledge and belief relies on a subtle
equivocation between an "evidence-in-hand" conception of knowledge and a weaker
"evidence-out-there" notion of what could come to be known. Our analysis leads
to a trimodal logic of knowledge, knowability, and belief interpreted in
topological subset spaces in which belief is definable in terms of knowledge
and knowability. We provide a sound and complete axiomatization for this logic
as well as its uni-modal belief fragment. We then consider weaker logics that
preserve suitable translations of Stalnaker's postulates, yet do not allow for
any reduction of belief. We propose novel topological semantics for these
irreducible notions of belief, generalizing our previous semantics, and provide
sound and complete axiomatizations for the corresponding logics.
