I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric.
There will be a metric space $X$ such that
If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not convergent in $Y$ for all, $Y$ such that $X\subset Y$ ?
I give some examples and the answer seems to be not, simply take $\overline{X}$, although
I'm not sure.
On the other hand, there is definition of cauchy sequence in topological spaces, if any, which would be the answer to the question?
I appreciate the patience and time for each of you, hope you can guide me
