TITLE: Is it possible to define Cauchy sequences in a topological space?
QUESTION [10 upvotes]: I know that we can define Cauchy sequences in topological vector spaces. How about in general topological spaces? Is it possible to define a Cauchy sequence in general topological spaces?

REPLY [8 votes]: 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.