[formula omitted]-sequential spaces and the images of metric spaces.

In this paper we introduce and study I s n -sequential spaces and I - snf -countable spaces. The following main results are obtained: let I be an ideal on N and X be a topological space, then (1) X is an I s n -sequential space if and only if each I s n -continuous (resp. quotient) mapping on the space X is continuous (resp. I s n -quotient). (2) X is an I -neighborhood space if and only if each I s n -quotient mapping on the space X is I -quotient, which answers [10, Question 4.12(2)] in the affirmative. (3) X is an I - snf -countable space if and only if X is a continuous and 1- I -covering image of a metric space. (4) If I is a K -uniform ideal on N , then each topological space is an I -neighborhood space. [ABSTRACT FROM AUTHOR]