Attention
This thread has been generalized to uniform spaces as general metric spaces.
Context
The context was the equivalence: $$K\text{ compact}\iff K\text{ totally bounded, complete}$$ That is a purely topological notion compared to a purely uniform notion and a hybrid notion.
Problem
Given a uniform structure $\mathcal{U}$.
Is there a way to define completeness intrinsically: $$\mathcal{U}\text{ complete}:\iff\ldots$$ The usual way is extrinsically via filters: $$\mathcal{U}\text{ complete}:\iff\left(\mathcal{F}:\mathcal{F}\text{ cauchy}\implies\mathcal{F}\text{ converges}\right)$$ (Note that filters are an order-theoretic notion!)