Limits in category theory and analysis

I think this doesn't quite work:

Let $mathcal{C}$ be the category whose objects are the point of $X$, and define
$$
mathrm{mor}_mathcal{C}(x,y) = { mbox{closed sets containing both $x$ and $y$} }.
$$
Composition is union.

Now (for example) a sequence ${ x_n}$ in $X$ defines a functor $F: mathbb{N} to mathcal{C}$ and a cone from $F$ to $y$ is essentially a single closed set
containing the entire sequence and $y$. Since this set must contain the topological limit $x$ of the sequence, this means that the cone factors through the same closed set viewed as a morphism $xto y$, so $x$ is the categorical colimit of $F$.

And since the morphism sets are symmetrical, the sequence ${ x_n}$ can be viewed as a contravariant functor $G: mathbb{N}to mathcal{C}$, and the topological limit $x$ is the categorical limit of $G$.

PROBLEM: the factorization is not unique!