Is it possible to define Cauchy sequences in a topological space?

Solution 1:

No. Consider $X=(0,1)$ and $Y=(1,\infty)$ equipped with the usual metric. These are homeomorphic as topological spaces, since the map $h:X\to Y$, defined by $$h(x)=\frac1x$$ is a homeomorphism. But $h$ maps the Cauchy sequence $a_n=\frac1n$ to $h(a_n)=n$, which is not a Cauchy sequence. So being a Cauchy sequence is not invariant under homeomorphisms, but depends on the choice of a metric.

Solution 2:

In general topological spaces Cauchy sequences are not defined. Let us think of a possible definition. In metric spaces, we all know the definition, and we could try to mimic it. However, what is the topological counterpart of "$d(p_n,p_m)<\varepsilon$"? We could try

Definition. A sequence $\{p_n\}_n$ is a Cauchy sequence if, for every open set $U$, there exists $N>1$ such that $p_n$ and $p_m$ belong to $U$ for all $n$, $m>N$.

But this definition does not mean that $p_n$ and $p_m$ are as "close" as we wish when $m$ and $m$ become large: already in $\mathbb{R}$, pick $U=(0,1) \cup (100,1000)$. What is required in the definition of Cauchy sequences is some kind of "uniform neighborhood". And indeed Cauchy sequences are defined in topological vector spaces and in topological uniform spaces.