In this paper we generalize the notion of I-continuity, which was defined in [V. Baláž, J.Červeňanský, T.Kostyrko, T.Šalát : I-convergence and I-continuity of real functions, Acta Mathematica, Faculty of Natural Sciences, Constantine the Philosopher University Nitra 5 (2002), 43-50] for real functions, to maps on topological spaces. We study the classes of topological spaces such that for each map on X I-continuity implies continuity.