If we consider a Cauchy sequence which is not convergent in a non complete metric space we see that the sequence wants to converge a point but it can not find its limit in the space. The divergence of sequence comes from the lack of the limit point. We face with this problem as the metric spaces are sequential spaces and in such spaces sequential closure is equivalent to the closure. My problem is that: can we go further away from this idea if we study in non complete uniform spaces, i.e. can we find a divergent Cauchy sequence which does not want to converge some limit points or which wants to converge to a point that is not in closure?
For example in the following example we see that we have a non complete uniform space. However, still the Cauchy sequence wants to converge somewhere outside but near the space.
Let $X$ be a non empty set and let $\mathcal{F}$ be the space of all $(0,1]$-valued functions on $X$. Consider the family of pseudo metrics on $\mathcal{F}$ which is given with $\mathcal{A}:=\left \{ p_{x}:x\in X\right \}$ where $p_{x}(f,g)=\left \vert f(x)-g(x)\right \vert $. The family $\mathcal{A}$ defines a uniformity on $\mathcal{F}$. The topology which is compatible with $(\mathcal{F},\mathcal{A})$ is the topology of pointwise convergence. I can see that $(\mathcal{F},\mathcal{A})$ is not complete if we choose $X=(0,1]$. But, still the Cauchy sequences which are not convergent want to converge to the zero function which is not in the space but very close to space.