Kuratowski convergence on compacta and Hausdorff metric convergence on compacta

This paper completes and improves results of P. Piccione and R. Sampalmieri [Comment. Math. Univ. Carolin. 36 (1995), no. 3, 551--562. Let (X, dX ), (Y, dY ) be two metric spaces and G be the space of all Y-valued continuous functions whose domain is a closed subset of X. If X is a locally compact metric space, then the Kuratowski convergence \tau_K and the Kuratowski convergence on compacta \tau_Kc coincide on G. Thus if X and Y are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology \tau_AW (generated by the box metric of dX and dY ) and \tau_Kc convergence on G, which improves the main result of the paper cited above. In the second part of paper we extend the definition of Hausdorff metric convergence on compacta for general metric spaces X and Y and we show that if X is locally compact metric space, then also \tau-convergence and Hausdorff metric convergence on compacta coincide in G.