I-Symmetric Spaces and the π-Images of Metric Spaces

Symmetric spaces and $sn$-symmetric spaces, as a generalization of metric spaces, have many important properties and have been widely discussed. We consider characterizations and mapping properties of $sn$-symmetric spaces under ideal convergence. $\mathcal{I}$-symmetric spaces and $\mathcal{I}$-$sn$-symmetric spaces are defined and studied. These not only generalize some classical results on symmetric spaces but also provide new directions to study generalized metric spaces using the notion of ideal convergence. As an application of $\mathcal{I}$-$sn$-symmetric spaces, some relevant properties of statistical convergence are obtained. Some unanswered questions in this field are raised.