Generalized convergence and generalized sequential spaces

We continue the study of g-convergence given in 2005 [Caldas, M.; Jafari, S. On $g$-US spaces. {\em Stud. Cercet. \c{S}tiin\c{t}. Ser. Mat. Univ. Bac\u{a}u} {\bf 14} (2004), 13--19 (2005).] by introducing the sequential $g$-closure operator and we prove that the product of $g$-sequential spaces is not $g$-sequential by giving an example. We further investigate sequential $g$-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 $g$-sequential implies sequential, $g$-Fr\'echet implies Fr\'echet and $g$-Fr\'echet implies $g$-sequential. However, the inverse conclusions are not true and some counter examples are given. Also, we show that strongly $g$-continuous image of a $g$-sequential space is $g$-sequential, if the map is quotient. Finally, we obtain a necessary and sufficient condition for a topological space to be $g$-sequential in terms of a sequentially $g$-quotient map.