TITLE: Duality between topology and bornology
QUESTION [18 upvotes]: I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:
Let $X$ be a set and let $\mathcal{P}(X)$ be the bounded lattice induced by the power set $P(X)$ together with unions, intersections, $X$ and $\emptyset$ as joints, meets, $1$ and $0$. An $\textbf{ideal}$ $\beta \subseteq \mathcal{P}(X)$ is called a $\textbf{bornology}$ if $\bigvee_{B \in \beta} \, B=1.$
This should be equivalent to the usual definition. An obvious dualization of this is a $\textbf{filter}$ $\nu \subseteq \mathcal{P}(X)$, s.t. $\bigwedge_{N \in \nu}=0$. 
This should be seen as the set of all neighbourhoods with respect to some topology on $X$ and the question is now how to extract a topology from it. I know that there is a unique topology for every neighbourhood system $\{N(x)\}_{x \in X}$ and of course I can define a trivial neighbourhood system from a given filter $\nu \subseteq \mathcal{P}(X)$ by setting $N(x)$ to be the subfilter of sets in $\nu$ containing $x$. This leads to the finest possible topology, but what I actually want is a unique coarsest neighbourhood system (s.t. $\nu = \cup_x \, N(x) $)! Unfortunately, I don't really have an idea how to start showing the existence of such a thing.. do you? Of course, any other idea how to define open sets is welcome! 
EDIT 1: it seems that coarsest is not the right property here, since for a given topology, the neighbourhoodfilter is in general not the coarsest neighbourhood system
EDIT 2: I guess I made a really stupid mistake here: the filter $\nu$ cannot distinguish between topologies $\tau,\tilde{\tau}$ with neighbourhood filters $N(x),\tilde{N}(x)$, s.t. $\bigcup_x \, N(x) =\bigcup_x \, \tilde{N}(x)= \nu$..
EDIT 3: For topological vector spaces $(X,\tau),(X,\tilde{\tau})$ it should be true that from $\bigcup_x \, N(x) =\bigcup_x \, \tilde{N}(x)$ follows $N(0)=\tilde{N}(0)$, thus $\tau = \tilde{\tau}$. I used both continuity of multiplication and addition.
Moreover, if $\tau$ is nontrivial, we have $\bigcap_{N \in \bigcup_x N(x)} \, N = \emptyset$. I didn't use that $\tau$ is Hausdorff.

REPLY [2 votes]: Since this query has resurfaced, I would like to suggest that the pro and ind categories of Grothendieck are a suitable framework to discuss the topic.  These are ways to extend a given categories by adding formally projective and inductive limits.
Let me start with the case of vector spaces since the situation here is more transparent and this is the one which has been studied in most detail, at least for bornologies.
If we start with the category of Banach spaces, we obtain respectively that of locally convex spaces and of convex bornological spaces (Buchwalter, Hogbe Nlend).  This suggests that the natural structure of a vector space (of functions, distributions) depends on whether they are natural projective or inductive limits of Banach spaces.  Thus the smooth functions on $I$, the unit interval—a lcs, the distributions thereon—a cbs.  Similarly, the continuous functions on the line—a lcs, the compactly supported measures thereon—a cbs.
These examples suggest continuing by incorporating duality in its functional analytic form.  The basic example is the functor which assigns to each Banach space its dual.  As a contravariant functor, this extends to one from the pro category into the ind category and vice versa, in our case form the category of lcs’s into cbs’s.  The fact that one can regard the dual of a lcs as a cbs (and vice versa) in a natural way has already been mentioned in the above answers and there are many indications external to category theory that is the more natural approach to duality rather than the spandrel of regarding the dual of a lcs as a lcs with various candidate topologies. (Health warning: this is very much a minority opinion).
Turning to the non linear case, the basic category is that of metric spaces with Lipschitz functions as morphisms. (In order to keep things simple, I will the blanket assumption that everything is sight is complete).  The corresponding pro category is that of uniform spaces.  The ind category has not, to my knowledge, been discussed but it can be 
defined directly as a class of bornologies, but enriched by the fact that
each bounded set is provided with a suitable metric.  Everybody has a collection of interesting uniform spaces in his toolbox,  most of the relevant bornological spaces have, in fact, suitable metrics.