A study of spaces and mappings in the sense of ideal convergence

Abstract

The ideal convergence of sequences in topological spaces not only includes the usual convergence of sequences, but also extends the statistical convergence of sequences with strong applying background. This paper discusses the subject of spaces and mappings in the sense of ideal convergence, and study the spaces defined by ideal convergence and how to represent them as the images of metric spaces under certain mappings. The following main results are obtained for an admissible ideal $\mathcal{I}$ on the set $\mathbb N$ of natural numbers.

(1)  A topological space $X$ is a seq-$\mathcal{I}$-space if and only if it is an $\mathcal{I}$-quotient image of a metric space.

(2) A topological space $X$ is a seq-$\mathcal{I}_{sn}$-space if and only if it is an $\mathcal{I}_{sn}$-quotient image of a metric space.

These show the unique role of $\mathcal{I}$-open sets and $\mathcal{I}_{sn}$-open sets in topological spaces, and present a version using the notion of ideals.