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)?
Best Answer
I believe this is true -- we can recover what the sequences converge to.
Say $(a_n)$ is a sequence in $X$ that we know converges, but we don't know what it converges to. There will be a unique $x \in X$ so that the new sequence $(a_1, x, a_2, x, a_3, x, a_4, x, \ldots)$ is convergent (which we can detect), and this $x$ is necessarily the limit of the sequence $(a_n)$.
Of course, this process is not "computable" in any sense. I have no idea how one would find such an $x$ in practice (though maybe it's doable in case $X$ is compact...) but, in the abstract, this shows that the convergent sequences remember limits.
I hope this helps ^_^