Generalized convergence and generalized sequential spaces

We continue the study of g-convergence given in 2005 [Caldas, M.; Jafari, S. On -US spaces. Stud. Cercet. Ştiinţ. Ser. Mat. Univ. Bacău 14 (2004), 13–19 (2005).] by introducing the sequential -closure operator and we prove that the product of -sequential spaces is not -sequential by giving an example. We further investigate sequential -continuity in topological spaces and present interesting theorems which are also new for the real case. It is shown that in a topological space the property of being -sequential implies sequential, -Fréchet implies Fréchet and -Fréchet implies -sequential. However, the inverse conclusions are not true and some counter examples are given. Also, we show that strongly -continuous image of a -sequential space is -sequential, if the map is quotient. Finally, we obtain a necessary and sufficient condition for a topological space to be -sequential in terms of a sequentially -quotient map.