Duality between topology and bornology

Question

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.

Answer 1

It was not clear to me at first what your question has to do with bornologies, but now (EDITED) I see it. Any collection $\nu$ of subsets of $X$ (I assume that $X$ is nonvoid through the remainder of the answer since $X=\varnothing$ has no filters) is a subbasis of a unique topology $\tau_\nu$ on $X$ - to wit, the intersection of all topologies on $X$ containing $\nu$. The fact that $\nu$ is a filter will ensure that $\tau_\nu=\nu\cup\{\varnothing\}$. More precisely:

• Since $\nu$ is closed under finite intersections, we conclude that $\nu$ is even a basis of $\tau_\nu$. This implies that $\tau_\nu$ is the unique topology on $X$ for which $\nu$ is a basis and thus the coarsest topology on $X$ containing $\nu$;

• Since $\nu$ is closed under taking supersets (which, by the way, entails the fact that $1=X\in\nu$, implicitly used above), we conclude that any nonvoid open subset $U\in\tau_\nu$, being an union of members of $\nu$, belongs itself to $\nu$. Therefore, $\nu\cup\{\varnothing\}$ satisfies all the axioms for a topology on $X$. Since $\tau_\nu$ is the coarsest topology on $X$ containing $\nu$, we must have $\tau_\nu=\nu\cup\{\varnothing\}$, as claimed.

The property $\bigcap_{U\in\nu}U=\varnothing$ dual to a bornology being a cover of $X$ amounts to $\tau_\nu$ being $T_1$ and $X$ having at least two points (EDIT: in a previous version of the answer, it was suggested that $\tau_\nu$ was in this case only $T_0$ at best), since it means that for all $p\in X$ there is a $B\in\beta$ such that $x\not\in B$ (if $X=\{p\}$ then necessarily $\nu=\{X\}$ and therefore this property fails. Likewise, a singleton set admits no bornology which covers it). Therefore, given $X\ni p_j\not\in B_j\in\nu$, $j=1,2$ with $p_1\neq p_2$, we have that $B'_j=(B_1\cap B_2)\cup\{p_j\}\in\nu$ for $j=1,2$ with $p_1\not\in B'_2$, $p_2\not\in B'_1$. Conversely, if $\tau_\nu$ is $T_1$ and $X$ has at least two points, this implies that for all $p\in X$ we have at least one $B\in\nu$ which does not contain $p$, as claimed. On the other hand, $\nu=$ the collection of all nonvoid open subsets of $X$ being closed under taking supersets is considerably stronger than just being closed under unions, so this $\tau_\nu$ may be a very fine and disconnected topology, as the extreme case of Stone duality for complete Boolean algebras shows.

Remark: proper filters must satisfy $\varnothing\not\in\nu$, so including $0=\varnothing$ to $\tau_\nu$ by hand is necessary.

If $X$ is a vector space (over $\mathbb{R}$ or $\mathbb{C}$) and $\beta$ is a convex vector bornology (i.e. closed under addition, scalar multiplication and absolutely convex hulls), there is a different way to define a topology on $X$ - namely, you adopt as your neighborhood filter of zero the collection of all $\beta$-bornivorous disks (i.e. absolutely convex subsets that absorb all members of $\beta$). This defines a locally convex topology on $X$ - the finest such one w.r.t. which the bounded subsets are precisely the members of $\beta$. However, it is clear that the vector space structure of $X$ plays an essential role in this case. Interestingly, such a topology is Hausdorff if and only if $\beta$ covers $X$, and it is well known that a vector space topology in $X$ is Hausdorff if and only if it is $T_1$ (in fact, it is even completely regular in this case).

The duality between (convex vector) bornology and (locally convex vector) topology acquires a deeper meaning in the theory of locally convex vector spaces, since compatible (locally convex vector) topologies on (topological) dual spaces are defined in terms of (convex vector) bornologies of the original spaces, and vice-versa - more precisely, the seminorms in the dual space $E'$ defining such a topology may be taken as supremum seminorms over the members of a (convex vector) bornology $\beta$ on the original space $E$, provided $\beta$ is compatible with the topology of $E$ (i.e. consisting of bounded subsets of $E$). This can be seen as an extension of the concept of compact-open topology in function spaces. Again, this topology is $T_1$ (hence Hausdorff) if and only if $\beta$ covers $E$, since the former amounts to the above seminorms separating points in $E'$. Conversely, equicontinuous subsets of $E'$ form a (convex vector) bornology therein for each (locally convex vector) topology on $E$ for which $E'$ as the topological dual of $E$. However, since any singleton set in $E'$ is trivially equicontinuous, such a bornology always covers $E'$. The book "Bornologies and Functional Analysis" by Henri Hogbe-Nlend (North-Holland, 1977) discusses such topics in depth.

Answer 2

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.