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