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A topological space is \emph{hereditarily $k$-irresolvable} if none of itssubspaces can be partitioned into $k$ dense subsets, We use this notion toprovide a topological semantics for a sequence of modal logics whose $n$-thmember K4$\mathbb{C}_n$ is characterised by validity in transitive Kripkeframes of circumference at most $n$. We show that under the interpretation ofthe modality $\Diamond$ as the derived set (of limit points) operation,K4$\mathbb{C}_n$ is characterised by validity in all spaces that arehereditarily $n+1$-irresolvable and have the T$_D$ separation property.  We also identify the extensions of K4$\mathbb{C}_n$ that result when theclass of spaces involved is restricted to those that are weakly scattered, orcrowded, or openly irresolvable, the latter meaning that every non-empty opensubspace is 2-irresolvable. Finally we give a topological semantics for K4M,where M is the McKinsey axiom.

Modal Logics of Some Hereditarily Irresolvable Spaces