  In this manuscript, we claim that the newly introduced $\mathcal{F}$-metric
spaces are Hausdorff and also first countable. Moreover, we assert that every
separable $\mathcal{F}$-metric space is second countable. Additionally, we
acquire some interesting fixed point results concerning altering distance
functions for contractive-type mappings and Kannan-type contractive mappings in
this exciting context. However, most of the findings are well-furnished by
several non-trivial numerical examples. Finally, we raise an open problem
regarding the metrizability of such kind of spaces.
