In this work we introduce the topological state derivative, a noval approach to treat PDE-constrained topology optimisation problems. This notion allows to deal with point perturbations as well as more general perturbations like smooth hypersurfaces in a similar way. Furthermore, we draw a connection from the topological state derivative to the asymptotic expansion of the state equation, which is usually derived using boundary layer correctors. Finally, we present numerical results based on these ideas.