Bornology

Introduction

Topology records the small: an open set is a neighbourhood of each of its points, and the structure is generated by the neighbourhoods of the points. A bornology records the large: it is a family of subsets of a set, called the bounded sets, that covers the set, is closed under finite unions, and is closed under the formation of subsets. The two structures are independent: a set can carry a bornology with no topology, a topology with no bornology, and the two together. The bornology of a metric space, given by the metrically bounded sets, is the standard example, and it is unchanged if the metric is replaced by a bornologically equivalent one, so the bornology is the part of the metric structure that survives when the small distances are forgotten. The topological spaces enter through the bornology of relatively compact sets, which is defined on a locally compact space, and the functional analysis of Part III enters through the bounded sets of a normed or, more generally, a bornological vector space, where the bornology is a piece of the linear structure and the bornological and the topological notions of boundedness are compared.

This article develops the axioms for a bornology, its bases and subbases, the bornologies attached to a metric space and to a locally compact space, the lattice of bornologies on a set, the bounded maps and the associated category, and the modification of a topology by a bornology. The bornology is the fourth of the covering structures of this Part, alongside the topology, the uniformity and the proximity, and it is the one that forgets the local information entirely and retains only the large-scale information; the comparison with the other three is made in the last section. The bounded sets of the functional analysis belong to Part III, where the norms and the limits are available, and the coarse and large-scale geometry of metric spaces belongs to the metric-geometry article written in parallel; both are deferred explicitly. No measure, integral, derivative or analytic limit is used, and no physics is invoked.

Bornologies

The Axioms

Definition. A bornology on a set $X$ is a family $\mathcal{B}$ of subsets of $X$ satisfying:

(B1) $\mathcal{B}$ covers $X$, that is, $X = \bigcup_{B \in \mathcal{B}} B$;

(B2) $\mathcal{B}$ is closed under finite unions: if $B_1, \ldots, B_n \in \mathcal{B}$ then $B_1 \cup \cdots \cup B_n \in \mathcal{B}$;

(B3) $\mathcal{B}$ is hereditary: if $B \in \mathcal{B}$ and $A \subseteq B$ then $A \in \mathcal{B}$.

The members of $\mathcal{B}$ are the bounded sets of the bornology, and a bornological set, or bornological space, is a pair $(X, \mathcal{B})$. A subset that is not bounded is unbounded, and $X$ is bounded if $X \in \mathcal{B}$.

In the presence of (B2) and (B3) the axiom (B1) says that $\mathcal{B}$ is an ideal of subsets of $X$, possibly the improper ideal $\mathcal{P}(X)$, and the empty set belongs to $\mathcal{B}$ by heredity. Thus a bornology is an ideal of subsets that covers the underlying set, and this is the whole of the definition.

Example (the finite bornology). On any set $X$ the family of finite subsets is a bornology, the finite bornology, and it is the smallest bornology on $X$; for an infinite set the set $X$ itself is unbounded, and for a finite set the finite bornology is the whole power set. The finite bornology is the one attached to the discrete metric of a set, in the sense of the next section.

Example (the maximal bornology). The whole power set $\mathcal{P}(X)$ is a bornology, the maximal or trivial bornology, in which every set is bounded; it is the largest bornology on $X$. Every bornology on $X$ lies between the finite and the maximal one, so the family of bornologies is a bounded lattice, as in the fourth section.

Example (the bornology of a metric space). Let $(X,d)$ be a metric space and let $$ \mathcal{B}_d = \{ B \subseteq X : \operatorname{diam}_d(B) < \infty \}, \qquad \operatorname{diam}_d(B) = \sup \{ d(x,y) : x,y \in B \} . $$ Then $\mathcal{B}_d$ is a bornology, the metric bornology, and it is the family of bounded sets of the metric in the usual sense. The metric bornology is unchanged when the metric is replaced by any metric that is bornologically equivalent to $d$, meaning that the identity is a bounded map in both directions; the truncated metric $\min(d,1)$ is not of this kind, since every set has diameter at most $1$ for it, so it induces the same topology as $d$ and the maximal bornology. The topology and the bornology of a metric space are thus independent data.

Example (the compact bornology). Let $X$ be a locally compact Hausdorff space and let $$ \mathcal{B}_c = \{ B \subseteq X : \overline{B} \text{ is compact} \} $$ be the family of relatively compact subsets. Then $\mathcal{B}_c$ is a bornology: finite unions of relatively compact sets are relatively compact, subsets are, and the local compactness ensures that every point has a relatively compact neighbourhood, so the family covers $X$. It is the compact bornology, and on a compact space it is the maximal bornology. For a non-locally-compact space the relatively compact sets need not cover $X$, and then they form only a bornological base, not a bornology; the next subsection isolates the notion.

Example (the countable bornology). On a set $X$ of cardinality at most $\aleph_0$ the countable subsets form a bornology; on an uncountable set the countable subsets fail to cover $X$, so they do not.

Bases and Subbases

Definition. A base for a bornology $\mathcal{B}$ on $X$ is a subfamily $\mathcal{B}_0 \subseteq \mathcal{B}$ such that every member of $\mathcal{B}$ is contained in a member of $\mathcal{B}_0$. A subbase is a family $\mathcal{S}$ of subsets of $X$ whose members' finite unions form a base; the bornology generated by $\mathcal{S}$ is the family of all subsets of finite unions of members of $\mathcal{S}$.

Proposition. Every subbase of a covering family generates a smallest bornology containing it, and the bornology generated by a base $\mathcal{B}_0$ consists exactly of the subsets contained in some member of $\mathcal{B}_0$.

Proof. Given a family $\mathcal{S}$ covering $X$, let $\mathcal{B}$ be the family of subsets of finite unions of members of $\mathcal{S}$. Then $\mathcal{B}$ contains $\mathcal{S}$, covers $X$, and is hereditary by construction; it is closed under finite unions because a union of subsets of finite unions of members of $\mathcal{S}$ is again a subset of a finite union of members of $\mathcal{S}$. Any bornology containing $\mathcal{S}$ contains every finite union by (B2) and every subset of a finite union by (B3), hence contains $\mathcal{B}$. The statement for bases is the special case in which $\mathcal{S}$ is closed under finite unions.

Definition. A bornology has countable type if it has a countable base. A bornology is of finite type when its base can be taken to consist of finite sets, which is the finite bornology.

Example. The metric bornology of $\mathbb{R}^n$ has the base of the closed balls $B(0,k)$ for $k \in \mathbb{N}$, so it has countable type; the compact bornology of a $\sigma$-compact locally compact space has the same countable base, and the compact bornology of a non-$\sigma$-compact locally compact space does not have countable type.

Proposition. A bornology has countable type if and only if it has a base consisting of the members of an increasing sequence $B_1 \subseteq B_2 \subseteq \cdots$ of bounded sets; and the metric bornology of a separable metric space has countable type, the balls of integer radius about the points of a countable dense set forming a countable base.

Proof. A countable base $(B_n)$ is converted into an increasing sequence by $B_1' = B_1$ and $B_{n+1}' = B_n' \cup B_{n+1}$, which is a base by (B2); conversely an increasing sequence is a countable base. For the metric statement, a bounded set $S$ of diameter $M$ lies in the ball $B(d, M+1)$ for every point $d$ of a countable dense set, and the family of all such balls is countable.

Bounded Maps and the Category of Bornological Sets

Definition. Let $(X, \mathcal{B})$ and $(Y, \mathcal{C})$ be bornological sets. A map $f : X \to Y$ is bounded, or bornological, if $f(B) \in \mathcal{C}$ for every $B \in \mathcal{B}$. A bijective bounded map whose inverse is bounded is a bornological isomorphism.

Proposition. The bornological sets with the bounded maps form a category; the finite bornologies give the category of sets with all maps; the maximal bornologies give the category of sets with all maps; and the forgetful functor to the category of sets is faithful.

Proof. The composition of bounded maps is bounded, since $(g \circ f)(B) = g(f(B))$ and induction on the definition; the identity is bounded. When every set is bounded every map is bounded, which gives the two extreme cases. The remaining statements are immediate.

Proposition. Let $X$ and $Y$ carry bornologies and let $X \times Y$ carry the bornology generated by the products of bounded sets. Then the projections are bounded, and a map $f : Z \to X \times Y$ is bounded if and only if its two components are bounded. The same statement holds for the coproduct $X \sqcup Y$ with the bornology generated by the images of the bounded sets of the two summands, with the injections bounded and the statement about the components of a map out of the coproduct.

Proof. The projections of a product of bounded sets are bounded sets by heredity, and a map is bounded into a product exactly when both components are, by the definition of the generated bornology; the arguments for the coproduct are dual and use the covering condition.

Example. The bounded maps between metric spaces are exactly the maps that send bounded sets to bounded sets; since a uniformly continuous map sends bounded sets to bounded sets, they are strictly more general than the uniformly continuous maps, and they are incomparable with the continuous maps. The map $\mathbb{R} \to \mathbb{R}$, $x \mapsto x^2$, is not bounded for the usual bornology, while $x \mapsto \sin(x^2)$ is bounded and not uniformly continuous, and $x \mapsto 1/x$ on $(0,1)$ is unbounded although it is continuous on its domain.

The Lattice of Bornologies

Theorem. The family of bornologies on a set $X$, ordered by inclusion, is a complete lattice. Its smallest element is the finite bornology, its largest is the maximal bornology, the infimum of a family is its intersection, and the supremum is the bornology generated by the union.

Proof. The intersection of bornologies is a bornology: it covers $X$ because every bornology does, and is closed under finite unions and subsets because each member is. The supremum is described by the generation of the preceding section, and the generated bornology is the smallest one containing the union. The finite and the maximal bornology are the extreme members, the finite one being contained in every bornology by (B2) and (B3) applied to the singletons.

Definition. A bornology $\mathcal{B}$ is finer than $\mathcal{C}$, and $\mathcal{C}$ is coarser than $\mathcal{B}$, when $\mathcal{B} \supseteq \mathcal{C}$; in that case the identity map $(X, \mathcal{B}) \to (X, \mathcal{C})$ is bounded. Two metrics on $X$ are bornologically equivalent when they induce the same bornology, equivalently when the identity is bounded in both directions for the two metric bornologies.

Proposition. Two metrics $d$ and $d'$ on $X$ are bornologically equivalent if and only if every $d$-bounded set is $d'$-bounded and conversely; this holds when $d' \leq \varphi \circ d$ and $d \leq \psi \circ d'$ for nondecreasing functions $\varphi, \psi : [0,\infty) \to [0,\infty)$ with $\varphi(0)=\psi(0)=0$. Uniformly equivalent metrics are bornologically equivalent, and the converse fails.

Proof. The first statement is the definition. For the second, a $d$-bounded set $B$ has $d(x,y) \leq M$ for all pairs, so $d'(x,y) \leq \varphi(M)$, which is finite; the converse is symmetric. For the failure of the converse, the metrics $d(x,y) = |x-y|$ and $d'(x,y) = \min(|x-y|,1)$ on $\mathbb{R}$ have the same bounded sets but are not uniformly equivalent, since the identity from $d'$ to $d$ is not uniformly continuous.

Bornologies and Topologies

The Modification of a Topology by a Bornology

Definition. Let $X$ carry a topology $\tau$ and a bornology $\mathcal{B}$. The bornological modification $\tau_{\mathcal{B}}$ of $\tau$ is the family of sets $O \subseteq X$ such that $O \cap B$ is open in the subspace $B$ for every $B \in \mathcal{B}$, where each bounded set carries the subspace topology.

Proposition. The bornological modification $\tau_{\mathcal{B}}$ is a topology on $X$, it is finer than $\tau$, and it is the finest topology on $X$ agreeing with $\tau$ on every bounded set. The assignment $\mathcal{B} \mapsto \tau_{\mathcal{B}}$ is antitone in the bornology: a finer bornology gives a coarser modification. If the bornology is the maximal one then $\tau_{\mathcal{B}} = \tau$, and if the topology is $T_1$ and the bornology is the finite one then $\tau_{\mathcal{B}}$ is the discrete topology, because a finite subspace of a $T_1$ space is discrete; the modification by the compact bornology of a locally compact space is the original topology.

Proof. The family $\tau_{\mathcal{B}}$ is closed under arbitrary unions and finite intersections, since these operations commute with the intersection with a fixed bounded set, so it is a topology; it contains $\tau$ because an open set meets every subspace in an open set; and if a topology $\sigma$ agrees with $\tau$ on every bounded set, then $O \cap B \in \sigma|_B = \tau|_B$ for every $O \in \sigma$ and every $B \in \mathcal{B}$, so $O \in \tau_{\mathcal{B}}$. Antitonicity is immediate from the definition, more bounded sets imposing more conditions. When the bornology is the maximal one the condition includes $B = X$ and gives $\tau_{\mathcal{B}} = \tau$; when it is the finite bornology of a $T_1$ space the condition is vacuous, a finite subspace being discrete, and $\tau_{\mathcal{B}}$ is the discrete topology. For the compact bornology of a locally compact space, let $O \in \tau_{\mathcal{B}}$ and $x \in O$; the point $x$ has an open relatively compact neighbourhood $V$, and $O \cap V$ is open in $V$, hence open in $X$, so it is a neighbourhood of $x$ contained in $O$, whence $O \in \tau$.

The Compact Bornology and the Topological Bornologies

Theorem. Let $X$ be a Hausdorff space. The family of subsets contained in a finite union of compact subsets of $X$ is a bornology, the bornology generated by the compact subsets, and it is exactly the family of subsets with compact closure. It covers $X$ exactly when $X$ is locally compact, and then it is the compact bornology of the relatively compact sets; on a compact space it is the maximal bornology, and on a $\sigma$-compact locally compact space it has countable type.

Proof. A finite union of compact subsets of a Hausdorff space is compact, so a subset of such a union has compact closure; conversely a set with compact closure is contained in that compact closure. The family of subsets of finite unions of compact sets is hereditary and closed under finite unions by the construction of a generated bornology, so it is a bornology, and it covers $X$ exactly when every point lies in a compact set, which is local compactness for a Hausdorff space. A compact space is its own compact subset, so every subset is bounded and the bornology is maximal, and on a $\sigma$-compact space a countable covering by compact sets is a countable base for the compact bornology.

Remark. The topological bornologies that occur in practice are the compact bornology and the bornology of the precompact sets of a uniform space, where a set is precompact when it is totally bounded as a uniform subspace. The two coincide for a locally compact uniform space; for a general uniform space the precompact sets form a bornology that covers the space exactly when the uniform space is totally bounded in the large, and their relation to the compact bornology is the standard comparison between compactness and total boundedness in Metric, Uniform and Complete Spaces.

Bornology and the Other Covering Structures

Remark. The four structures carried by a set in this Part — topology, uniformity, proximity and bornology — are independent, and the bornology is the one that sees only the large scale. A topology is generated by its neighbourhoods and a uniformity by its entourages, both of which are local data; a proximity records the nearness of pairs of sets, which is intermediate; a bornology is generated by the bounded sets, which are the sets that are small only in the sense of being contained in a member of the family, with no reference to the points at all. A subset of a metric space is bounded in the space exactly when it is bounded in the completion, and the compact bornology of a locally compact space is unchanged by passing to the one-point compactification except for the added point, so the bornological data are exactly the data invariant under the completion procedures of the earlier articles.

Remark. The bounded maps between metric spaces are the morphisms of the large-scale geometry: two metric spaces are bornologically isomorphic when there is a bijection that is bounded in both directions, and the bornology is the coarsest structure under which the bounded maps are the morphisms. The finer large-scale structures — the coarse structures, whose entourages are families of subsets of $X \times X$ subject to a coarse properness condition, and the Lipschitz and the uniform structures — refine the bornology, and they belong, where the distance is the primary structure and the bornology is one of its derived objects. The passage from a bornology to a large-scale structure requires additional choices, and records them.

The Bounded Sets of Part III

Remark. In a normed space the metric bornology is the family of sets of finite diameter, equivalently of sets contained in a ball, and the bornology is the part of the normed structure that is visible at large distances. A bounded linear map between normed spaces is bounded as a map of bornological sets, and conversely a linear map that sends bounded sets to bounded sets is bounded in the operator sense, so for linear maps the two notions of boundedness agree; for nonlinear maps they do not, as the example $x \mapsto x^2$ on $\mathbb{R}$ shows. The bornological spaces of functional analysis are the locally convex spaces whose topology is determined by their bounded sets, in the sense that every seminorm bounded on the bounded sets is continuous; this is a property of the topological vector space, it uses the seminorms and the completeness, and it belongs to Part III, where the normed and the topological vector spaces are constructed. The present article stops at the bornology of a set and at the comparison with the topology, and it records only that the bornology is the part of the normed structure that the functional-analytic theory of boundedness uses.

Summary

A bornology on a set $X$ is a family of subsets, the bounded sets, that covers $X$, is closed under finite unions and is hereditary; equivalently, it is an ideal of subsets that covers $X$. The finite bornology is the smallest and the maximal bornology $\mathcal{P}(X)$ the largest; a metric space carries the metric bornology of the sets of finite diameter; a locally compact Hausdorff space carries the compact bornology of the relatively compact sets; a uniform space carries the bornology of the precompact sets. A base is a cofinal subfamily, the bornology generated by a subbase consists of the subsets of the finite unions of its members, and a bornology has countable type when it has a countable base. The bounded maps are the maps carrying bounded sets to bounded sets; they form a category, in which the product and the coproduct carry the bornologies generated by the products and the images of the bounded sets; and the bornological sets are the objects of the large-scale geometry. The bornologies on a set form a complete lattice under inclusion, with intersection as infimum and generation by the union as supremum; two metrics are bornologically equivalent when they have the same bounded sets, which is weaker than uniform equivalence. A topology is modified by a bornology to the finest topology agreeing with it on every bounded set, a modification that is finer than the original and becomes coarser as the bornology becomes finer, and the four covering structures of this Part — topology, uniformity, proximity and bornology — are independent, the bornology being the one that forgets the local information. The bornological vector spaces and the bounded linear maps of the functional analysis belong to Part III, where the norms and the limits are available.

Summary of Notation

Symbol	Meaning
$(X, \mathcal{B})$	Bornological set; $\mathcal{B}$ the family of bounded sets
(B1), (B2), (B3)	Covering, finite unions, heredity
$\mathcal{B}_d$	Metric bornology; sets of finite $d$-diameter
$\mathcal{B}_c$	Compact bornology; relatively compact sets
relatively compact	Subset with compact closure
base, subbase	Cofinal subfamily; generating family
countable type	Existence of a countable base
bounded map	$B \in \mathcal{B} \Rightarrow f(B) \in \mathcal{C}$
bornological isomorphism	Bijection bounded in both directions
finer, coarser	$\mathcal{B} \supseteq \mathcal{C}$; identity is bounded
bornologically equivalent metrics	Same family of bounded sets
$\tau_{\mathcal{B}}$	Bornological modification of the topology $\tau$
precompact subset	Totally bounded uniform subspace
$\operatorname{diam}_d(B)$	Diameter of $B$ in the metric $d$