I am aware that in a first countable space (and thus any metric space) is completely determined by its convergent sequences and their limits, i.e.,
If $\tau_1$ and $\tau_2$ are two first countable topologies on a set $X$ such that $x_i\to c$ in $\tau_1$ iff $x_i\to c$ in $\tau_2$, then $\tau_1 = \tau_2$.
However, this raises the following question: If two metrics on a space have the same convergent sequences, will they have the same limits as well (and thus the same topology)?
