In this manuscript, we prove that the newly introduced F-metric spaces are Hausdorff and first countable. We investigate some interrelations among the Lindelöfness, separability and second countability axiom in the setting of F-metric spaces. Moreover, we acquire some interesting fixed point results concerning altering distance functions for contractive type mappings and Kannan type contractive mappings in this exciting context. In addition, most of the findings are well-furnished by several non-trivial examples. Finally, we raise an open problem regarding the structure of an open set in this setting,