nLab
proximity space

Idea

In addition to the well-known topological spaces, many other structures can be used to found topological reasoning on sets, including uniform spaces and proximity spaces. Proximity spaces provide a level of structure in between topologies and uniformities; in fact a proximity is equivalent to an equivalence class of uniformities with the same totally bounded reflection.

Proximity spaces are often called nearness spaces, but this term has other meanings in the literature. One can clarify with the term set–set nearness space.

Definition

A proximity structure (or set–set nearness structure) on a set $X$, or a proximity relation (or nearness relation) on the power set $P(X)$ of subsets of $X$, is a binary relation $\delta$ on $P(X)$ such that

1. symmetry: $A\delta B$ iff $B\delta A$;

2. binary additivity: $A\delta B\cup C$ iff either $A\delta B$ or $A\delta C$;

3. nullary additivity: it is never true that $A\delta \varnothing$;

4. $\{x\}\delta \{y\}$ if $x=y$

5. if for every $C,D\subset X$ such that $C\cup D=X$, either $A\delta C$ or $B\delta D$, then $A\delta B$.

Another axiom one may require is the converse of (4):

• separation: $x=y$ if $\{x\}\delta \{y\}$

In general, we say that $A$ and $B$ are proximate (or near) if $A\delta B$, and apart otherwise. We also write $A\ll B$ if $(X\setminus A)\delta B$.

A proximity space (or set–set nearness space) is a set $X$ equipped with a proximity structure $\delta$. The proximity structure or proximity space is separated if it satisfies the separation axiom (the converse of 4); note that many authors require this by default.

Variations

There are many variations possible in the list of axioms; one important consequence of the above (sometimes listed separately, allowing additivity to be weakened) is this:

• isotony: if $A\subset C$ and $B\subset D$, then $C\delta D$ if $A\delta B$.

It is also possible to write the definition in terms of the apartness relation or the relation $\ll$. In particular, a (set–set) apartness space is a set $X$ equipped with a binary relation $\bowtie$ on $P(X)$ such that the negation of $\bowtie$ is a proximity relation. This is the preferred formulation in constructive mathematics (although you'll want to rephrase the definition axiom by axiom to remove spurious double negations).

The category $\mathrm{Prox}$

If $X$ and $Y$ are proximity spaces, then a function $f:X\to Y$ is said to be proximally continuous if $A\delta B$ implies $f(A)\delta f(B)$. In this way we obtain a category $\mathrm{Prox}$, whose evident forgetful functor $\mathrm{Prox}\to \mathrm{Set}$ makes it into a topological concrete category.

Relation to other topological structures

Topological spaces

Every proximity space is a topological space; let $x$ belong to the closure of $A\subset X$ iff $\{x\}\delta A$. This topology is always completely regular, and Hausdorff (hence Tychonoff) iff the proximity space is separated; see separation axiom. Proximally continuous functions are continuous for the induced topologies, so we have a functor $\mathrm{Prox}\to \mathrm{Top}$ over $\mathrm{Set}$.

Conversely, if $(X,\tau )$ is a completely regular topological space, then for any $A,B\subset X$ let $A\delta B$ iff $A\ne \varnothing \ne B$ and there is no continuous function $f:X\to I=[0,1]$ such that $f(x)=0$ on $A$ and $f(x)=1$ on $B$. This defines a proximity structure on $X$, which induces the topology $\tau$ on $X$, and which is separated iff $\tau$ is a Hausdorff (hence Tychonoff) topology.

In general, a completely regular topology may be induced by more than one proximity. However, if it is moreover compact, then it has a unique compatible proximity.

Uniform spaces

If $U$ is a uniformity on $Y$ (making it into a uniform space), then for all $A,B\subset Y$ let $A\delta B$ iff $V\cap (A\times B)\ne \varnothing$ for every entourage (aka vicinity) $V\in U$. This also defines a proximity structure on $Y$.

Uniformly continuous functions are proximally continuous for the induced proximities, so we have a functor $\mathrm{Unif}\to \mathrm{Prox}$ over $\mathrm{Set}$. Moreover, the composite $\mathrm{Unif}\to \mathrm{Prox}\to \mathrm{Top}$ is the usual “underlying topology” functor of a uniform space, i.e. the topology induced by the uniformity and the topology induced by the proximity structure are the same.

Conversely, if $X$ is a proximity space, consider the family of sets of the form

$$\bigcup _{k=1}^n{A}_k\times A_k$$\bigcup_{k=1}^n A_k \times A_k

where $(A_k)$ is a finite family of sets such that there exists a finite family of sets $(B_k)$ with $B_k\ll A_k$ and $X={\bigcup }_{k=1}^n{B}_k$. These sets form a base for a totally bounded uniformity on $X$, which induces the given proximity.

In fact, this is the unique totally bounded uniformity which induces the given proximity: every proximity-class of uniformities contains a unique totally bounded member. Moreover, a proximally continuous function between uniform spaces with totally bounded codomain is automatically uniformly continuous. Therefore, the forgetful functor $\mathrm{Unif}\to \mathrm{Prox}$ is a left adjoint, whose right adjoint also lives over $\mathrm{Set}$, is fully faithful, and has its essential image given by the totally bounded uniform spaces.

In general, proximally continuous functions need not be uniformly continuous, but in addition to total boundedness of the codomain, a different sufficient condition is that the domain be a metric space.

Syntopogenous spaces

A proximity space can be identified with a syntopogenous space which is both simple and symmetric; see syntopogenous space.

Compactifications

The (separated) proximities inducing a given (Hausdorff) completely regular topology can be identified with (Hausdorff) compactifications of that topology. The compactification corresponding to a proximity on $X$ is called its Smirnov compactification. The points of this compactification can be taken to be clusters in $X$, which are defined to be collections $\sigma$ of subsets of $X$ such that

1. If $A\in \sigma$ and $B\in \sigma$, then $A\delta B$.
2. If $A\delta C$ for all $C\in \sigma$, then $A\in \sigma$.
3. If $(A\cup B)\in \sigma$, then $A\in \sigma$ or $B\in \sigma$.

Related concepts

• uniform space, syntopogenous space

References

• R. Engelking, General topology, chapter 8.
• Douglas Bridges et al, Apartness, topology, and uniformity: a constructive view, pdf
• S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge University Press 1970