Introduction
This may seem like a weird question or even a silly one, but topology is vast and I find I make quicker progress working my way through it by trying ideas out loud within earshot of people that can cut them down rapidly if I'm wasting my time. It may seem more like a discussion, but I would prefer it took place as an actual post because I want to do more than just chat about it. There is a specific question at the end of this.
Main Event
Starting at the bottom (or top if you like) of the abstract analysis ladder we have a set with no structure. Then we go up (or down) to the next rung and define a topology and we have continuity of a function $f: X \to Y$, $X$ and $Y$ both topological spaces, with $U = f^{-1}(V)$ open in $X$ whenever $V$ open in $Y$. We can already talk about the continuity of addition and scalar multiplication in a vector space, just by defining a topology. With limits of sequences it involves having arbitrary open neighbourhoods of a point leaving out only finitely many points of the sequence.
Then, if we want things to have distances between them, we move to another rung and define a metric, preferably one of actual use in solving an interesting problem in analysis as opposed to a pathological counterexample, and determine if our spaces of interest are metrizable and so on.
The Actual Question
Now we can start talking about complete metric spaces. Every where I look, completeness is discussed in the context of a metric space. For example, in Steen and Seebach's Counterexamples in Topolgy, their definition of completeness is given in terms of a metric space, Munkres likewise. Here's where I muse that we've made topological versions of continuity and convergence that can be applied just using a topology, so is there a version of completeness that can be defined purely topologically? Convergence can, so if Cauchy could then there it would be.
Philosophical Epilogue
For me the whole point of building the current massive edifice of abstract mathematical machinery was to illuminate everything at the most fundamental level in order to refine our understanding and move on to greater mathematical heights rather than get bogged down in the particular details of a particular version of a particular type of problem. Are we succeeding? Or is maths just becoming far too complicated?
Conclusion
After the discussion below, the point highlighted is that Cauchyness involves closeness and how to do that without a metric. I had originally mistakenly thought Cauchyness could find its way into a pure topological setting because I had forgotten that I was using a topological definition that was being used on vector spaces without the advantage of a metric. The advantage of subtracting is none the less there and kills the point. I realised how careful one has to be with the words one uses to discuss this, even in matters that are allowed. What does "zero" even mean in an arbitrary set?
Originally above I used the word "small" when referring to convergence. It's kind of implied in the fact that $N$ may have to be really, really big depending on which open neighbourhood you choose, but we have no way of measuring "small" at the pure topological stage. Anyway, the answer is clear, there has to be something more than just the definition of a topology to be able to discuss completion.
Even the idea of compactness doesn't get us around it in general. In spaces where the unit ball (already measuring) is not compact, we need equi-continuity to establish the compactness.
I notice also that nobody wanted to get in to the philosophical question.
