We introduce and develop a topological semantics of conservativity logics and interpretability logics. We prove the topological compactness theorem of consistent normal extensions of the conservativity logic $\mathbf{ CL}$ by extending Shehtman's ultrabouquet construction method to our framework. As a consequence, we prove that several extensions of $\mathbf{ CL}$ such as $\mathbf{ IL}$, $\mathbf{ ILM}$, $\mathbf{ ILP}$ and $\mathbf{ ILW}$ are strongly complete with respect to our topological semantics.
Categories: math.LO