In this paper we study a simple, purely topological, cellular model which is allowed to evolve through a Glauber-Kawasaki process. We find a non-thermodynamic transition to a glassy phase in which the energy (defined as the square of the local cell topological charge) fails to reach the equilibrium value below a characteristic temperature which is dependent on the cooling rate. We investigate a correlation function which exhibits ageing behaviour, and follows a master curve in the stationary regime when time is rescaled by a factor of the relaxation time t_r. This master curve can be fitted by a von Schweidler law in the late β-relaxation regime. The relaxation times can be well fitted at all temperatures by an offset Arrhenius law. A power law can be fitted to an intermediate-temperature regime; the exponent of the power law and the von Schweidler law roughly agree with the relationship predicted by mode-coupling theory. By defining a suitable response function, we find that the fluctuation-dissipation ratio is held until sometime later than the appearance of the plateaux; non-monotonicity of the response is observed after this ratio is broken, a feature which has been observed in other models with dynamics involving activated processes.