TITLE: What sequences are Cauchy in all metrics for a given topology?
QUESTION [11 upvotes]: Different metrics for the same topology can have different sets of Cauchy sequences.  But I'm interested in what sequences are Cauchy in every metric for a given topology.  For a completely metrizable topology, the answer is obvious: the convergent sequences.  (Where convergence is a property of the topology independent of the metric.)
But my question is, if a topology is metrizable but not completely metrizable (like $\mathbb{Q}$ with the standard topology), is it possible for non-convergent sequences to be Cauchy in every metric for a given topology?

REPLY [2 votes]: Here is an alternate proof that a non-convergent sequence is always non-Cauchy in some metric.  Let $(X,d)$ be a metric space and $(x_n)$ be a Cauchy but non-convergent sequence in $X$.  Note that the set $A=\{x_n\}$ is closed and discrete in $X$, since any accumulation point of $A$ would be a limit of $(x_n)$ (since it is Cauchy) but $(x_n)$ does not converge.  Now let $f:A\to\mathbb{N}$ be a bijection ($A$ must be infinite or else $(x_n)$ would converge).  By the Tietze extension theorem, $f$ extends to a continuous function $g:X\to\mathbb{R}$, which has the property that $g(x_n)\to\infty$.
We can now define a new metric $d'$ by $d'(x,y)=d(x,y)+|g(x)-g(y)|$.  Since $g$ is continuous, $d'$ induces the same topology as $d$.  Since $g(x_n)\to\infty$, $(x_n)$ is not Cauchy with respect to $d'$.