Set convergence and local connectedness of approximate limit

We investigate the local connectivity of the approximate limit by means of the convergence of sets in the product. We show that the projections are monotonous, provided the spaces are normal, the product is (A)-compact and the bonding maps are monotonous. An example shows that the local connectivity of the limit of the approximate inverse system is necessary in order that the projections be monotonous whenever the bonding maps are.