If two metric spaces are homeomorphic then out, we can say convergent sequences are 'similar' in nature , Could we comment on the nature of cauchy sequences in homeomorphic metric spaces?
For instance
The space of natural numbers with usual metric $\mathbb{N}$ and $\mathbb{M} = \{\frac{1}{n} | n\in \mathbb{N} , n\geq1\}$ are homeomorphic with the map $f(x)=\frac{1}{x}$.
In both spaces convergent sequences are eventually constant sequences, but in $\mathbb{N}$ all cauchy sequences are eventually constant sequences hence convergent, but in $\mathbb{M}$ consider the sequence $x_n = 1/n , n\geq1$ this sequence is cauchy but not convergent to any point in $\mathbb{M}$.