CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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. (See for example this article.) One can clarify with the term set–set nearness space. The same goes for the term apartness space, which is another way to look at the same basic idea.
A proximity space is a kind of structured set: it consists of a set $X$ (the set of points of the space) and a proximity structure on $X$. This proximity structure is given by any of various binary relations between the subsets of $X$:
The conditions required of these relations are given below in the Definitions. (We say ‘proximal neighbourhood’ instead of simply ‘neighbourhood’ to avoid misapplying intuition from general topology. That $A \ll B$ doesn't necessarily mean that $A \subseteq Int(B)$; sometimes it means that $Cl(A) \subseteq Int(B)$, which is stronger, or something else.)
In classical mathematics, these relations are all interdefinable:
(Here, $\neg$ indicates negation of truth values, while $B'$ is the complement of $B$ in $X$.)
In constructive mathematics, any one of these relations may be taken as primary and the others defined using it; thus we distinguish, constructively, between a set–set nearness space, a set–set apartness space, and a set–set neighbourhood space.
From the previous section, we have a set $X$ and we are discussing binary relations $\delta, \bowtie, \ll$ on $X$. These are required to satisfy the following conditions; in each row, the conditions for the various relations are all equivalent (classically). In these conditions, $x, y$ are points, while $A, B, C$ are subsets, and we require them for all points or subsets. We list the conditions roughly in order of increasing optional-ness, then define terminology for relations satisfying them.
Name | Condition for nearness | Condition for apartness | Condition for proximal neighbourhoods |
---|---|---|---|
Isotony (left) | If $A \supseteq B \;\delta\; C$, then $A \;\delta\; C$ | If $A \subseteq B \bowtie C$, then $A \bowtie C$ | If $A \subseteq B \ll C$, then $A \ll C$ |
Isotony (right) | If $B \;\delta\; C \subseteq D$, then $B \;\delta\; D$ | If $B \bowtie C \supseteq D$, then $B \bowtie D$ | If $B \ll C \subseteq D$, then $B \ll D$ |
Additivity (left, nullary) | It is false that $\emptyset \;\delta\; A$ | $\emptyset \bowtie A$ | $\emptyset \ll A$ |
Additivity (right, nullary) | It is false that $A \;\delta\; \emptyset$ | $A \bowtie \emptyset$ | $A \ll X$ |
Additivity (left, binary) | If $A \cup B \;\delta\; C$, then $A \;\delta\; C$ or $B \;\delta\; C$ | If $A \bowtie C$ and $B \bowtie C$, then $A \cup B \bowtie C$ | If $A \ll C$ and $B \ll C$, then $A \cup B \ll C$ |
Additivity (right, binary) | If $A \;\delta\; B \cup C$, then $A \;\delta\; B$ or $A \;\delta\; C$ | If $A \bowtie B$ and $A \bowtie C$, then $A \bowtie B \cup C$ | If $A \ll B$ and $A \ll C$, then $A \ll B \cap C$ |
Reflexivity (general) | If $A$ meets $B$ (their intersection is inhabited), then $A \;\delta\; B$ | If $A \bowtie B$, then $A$ and $B$ are disjoint | If $A \ll B$, then $A \subseteq B$ |
Reflexivity (for singletons) | $\{x\} \;\delta\; \{x\}$ | It is false that $\{x\} \bowtie \{x\}$ | If $\{x\} \ll A$, then $x \in A$ |
Normality (constructive) | If for every $D, E \subseteq X$ such that $D \cup E = X$, either $A \;\delta\; D$ or $E \;\delta\; B$, then $A \;\delta\; B$ | If $A \bowtie B$, then for some $D, E \subseteq X$ such that $D \cup E = X$, both $A \bowtie D$ and $E \bowtie B$ | If $A \ll B$, then for some $D, E \subseteq X$ such that $D \subseteq E$, both $A \ll D$ and $E \ll B$ |
Normality (simplified) | If for every $D \subseteq X$, either $A \;\delta\; D$ or $D' \;\delta\; B$, then $A \;\delta\; B$ | If $A \bowtie B$, then for some $D \subseteq X$, both $A \bowtie D$ and $D' \bowtie B$ | If $A \ll B$, then for some $D \subseteq X$, both $A \ll D$ and $D \ll B$ |
Symmetry (constructive) | $A \;\delta\; B$ iff $B \;\delta\; A$ | $A \bowtie B$ iff $B \bowtie A$ | If $A \ll B$, $A \cup C = X$, and $B \cap D = \empty$, then $D \ll C$ |
Symmetry (simplified) | $A \;\delta\; B$ iff $B \;\delta\; A$ | $A \bowtie B$ iff $B \bowtie A$ | $A \ll B$ iff $B' \ll A'$ |
Separation | If $\{x\} \delta \{y\}$, then $x = y$ | Unless $\{x\} \bowtie \{y\}$, then $x = y$ | $x = y$ if, for all $A$, $y \in A$ whenever $\{x\} \ll A$ |
Perfection (left) | If $A \;\delta\; B$, then $\{x\} \;\delta\; B$ for some $x \in A$ | $A \bowtie B$ if $\{x\} \bowtie B$ for all $x \in A$ | $A \ll B$ if $\{x\} \ll B$ for all $x \in A$ |
Perfection (right) | If $A \;\delta\; B$, then $A \;\delta\; \{y\}$ for some $y \in B$ | If $A \bowtie \{y\}$ for all $y \in B$, then $A \bowtie B$ | If (for all $y$) $y \in B$ if (for all $C$) $y \in C$ if $A \ll C$, then $A \ll B$ |
When both left and right rules are shown, we only need one of them if we have Symmetry, but we need both if we lack Symmetry (or if we are using proximal neighbourhoods in constructive mathematics). Even so, Isotony is usually given on both sides, since it is convenient to combine both directions into a single statement. On the other hand, Isotony is equivalent to the converse of binary Additivity, so sometimes these are combined instead (so Isotony does not explicitly appear), usually on only one side when Symmetry is used.
Whether made explicit or not, Isotony is very fundamental, and it is what allows the axioms after Additivity to be written in different forms. In particular, we need Reflexivity only for singletons, although this is often not done (to avoid mentioning points). Similarly, we usually simplify Normality as shown (although this is appropriate for constructive mathematics only when defining neighbourhood spaces). In the same vein, Symmetry for proximal neighbourhoods is usually given in the simplified form (although now that is not appropriate for constructive mathematics).
A topogeny is a relation that satisfies both forms of Isotony and all four forms of Additivity. A quasiproximity is a topogeny that also satisfies Reflexivity and Normality. A topogeny (or quasiproximity) is symmetric if it satisfies Symmetry; a proximity is a symmetric quasiproximity. A topogeny or (quasi)-proximity is separated if it satisfies Separation. A topogeny or quasiproximity is perfect if it satisfies left Perfection, coperfect if it satisfies right Perfection, and biperfect if it satisfies both; a proximity (or a symmetric topogeny) is usually called simply perfect if it satisfies any form of Perfection, because then it must satisfy both (except in constructive mathematics using proximal neighbourhoods).
A (quasi)-proximity space is a set equipped with a (quasi)-proximity. All of these terms may be used with nearness, apartness, or proximal neighbourhoods, as explained in the previous section; nearness is usually the default.
Some authors require a (quasi)-proximity to be separated; conversely, some authors do not require a (quasi)-proximity to satisfy Normality. The term ‘topogeny’ is also not found in the literature (except here, in a generalization of nearness spaces); it is derived from ‘topogenous relation’, a term used in the theory of syntopogenous spaces for a nearness topogeny satisfying Reflexivity. (Thus, quasiproximities and topogenous relations are the same thing for authors who use nearness and do not require Normality.) The terminology for Perfection also comes from syntopogenous spaces.
If $X$ and $Y$ are (quasi)-proximity spaces, then a function $f: X \to Y$ is said to be proximally continuous if $A \;\delta\; B$ implies $f_*(A) \;\delta\; f_*(B)$, equivalently if $A \bowtie B$ whenever $f_*(A) \bowtie f_*(B)$, equivalently if $f^*(C) \ll f^*(D)$ whenever $C \ll D$. In this way we obtain categories $QProx$ and $Prox$; the forgetful functors $QProx \to Set$ and $Prox \to Set$ (taking a space to its set of points) make them into topological concrete categories.
Given points $x, y$ of a (quasi)-proximity space, let $x \leq y$ mean that $x$ belongs to every proximal neighbourhood of $\{y\}$, or equivalently (via Isotony) that $\{y\} \;\delta\; \{x\}$. By Reflexivity, $\leq$ is reflexive; by Normality, $\leq$ is transitive. (In fact, we can use these to deduce that $x \leq y$ iff every proximal neighbourhood of $\{y\}$ is a proximal neighbourhood of $\{x\}$, which is manifestly reflexive and transitive.) Therefore, $\leq$ is a preorder.
If the quasiproximity satisfies Symmetry, then this preorder is symmetric and hence an equivalence relation.
Regardless of Symmetry, a (quasi)-proximity space is separated iff this preorder is the equality relation. That is, $x = y$ if $x$ belongs to every proximal neighbourhood of $\{y\}$, or equivalently if every proximal neighbourhood of $\{y\}$ is a proximal neighbourhood of $\{x\}$, or equivalently if $\{x\}$ is near $\{y\}$, or equivalently if $\{x\}$ is not apart from $\{y\}$. This may be viewed as a converse of simplified Reflexivity, which states that $\{x\} \;\delta\; \{y\}$ whenever $x = y$.
Conversely, given a set equipped with a preorder $\leq$, let $A \;\delta\; B$ if $x \leq y$ for some $x \in A$ and some $y \in B$, or equivalently let $A \bowtie B$ if $x \leq y$ for no $x \in A$ and no $y \in B$, or equivalently let $A \ll B$ if $x \leq y$ for $x \in A$ implies $y \in B$. Then we have a quasiproximity space which is symmetric iff $\leq$ is.
In this way, we get the category $Proset$ of preordered sets as a reflexive subcategory of $QProx$, with the category $Setoid$ of setoids (sets equipped with equivalence relations) as a reflexive subcatgory of $Prox$.
Every proximity space is a topological space; let $x$ belong to the closure of $A \subseteq X$ iff $\{x\} \;\delta\; A$, or equivalently let $x$ belong to the interior of $A$ iff $\{x\} \ll A$. 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 $Prox \to Top$ over $Set$.
Conversely, if $(X,\tau)$ is a completely regular topological space, then for any $A, B \subseteq X$ let $A \bowtie B$ iff there is a continuous function $f: X\to [0,1]$ such that $f(x) = 0$ for $x \in A$ and $f(x) = 1$ for $x \in B$. 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, given above. In the case of a compact Hausdorff space (or more generally any normal regular space), we then have $A \ll B$ iff $Cl(A) \subseteq Int(B)$.
If $U$ is a uniformity on $Y$ (making it into a uniform space), then for all $A, B \subseteq Y$ let $A \;\delta\; B$ iff $V \cap (A \times B)$ is inhabited for every entourage (aka vicinity) $V$. This also defines a proximity structure on $Y$.
Uniformly continuous functions are proximally continuous for the induced proximities, so we have a functor $Unif \to Prox$ over $Set$. Moreover, the composite $Unif \to Prox \to Top$ 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
where $(A_k)_k$ is a list (a finite family) of sets such that there exists a same-length list of sets $(B_k)_k$ 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 $Unif \to Prox$ is a left adjoint, whose right adjoint also lives over $Set$, 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.
A proximity space can be identified with a syntopogenous space which is both simple and symmetric; see syntopogenous space.
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
As a poset, the power set $\mathcal{P}X$ of $X$ may be regarded as a category enriched over truth values. There is a notion of a bimodule over a category, also called (more specifically) a distributor or profunctor.
Then a profunctor from $\mathcal{P}X$ to itself is precisely a binary relation $\ll$ on subsets of $X$ that satisfies Isotony. Adding Reflexivity makes it a co-pointed profunctor, and Normality morally makes it a coassociative coalgebra with Reflexivity as counit. (Actually, coassociativity is trivial when enriched over truth values, as is the claim that Reflexivity, once it exists, is a counit, but we say ‘coassociative’ to clarify which sense of ‘coalgebra’ we mean.)
The sense in which Normality makes this a coalgebra is actually a bit involved, and it only quite works because of Additivity. A coalgebra with a given profunctor $\ll$ as its underlying bimodule has the structure of an operation that, given $x \ll z$, takes this to an equivalence class of $y$ such that $x \ll y \ll z$, where $y$ is equivalent to $y'$ if $y \subseteq y'$ (or by any equivalence that follows). By Isotony and left binary Additivity, $x \ll y \cup y' \ll z$ (or use right Additivity and $y \cap y'$); since $y, y' \subseteq y \cup y'$, we have the desired equivalence.
This suggests that if we want a notion of proximity without Additivity, then Normality must become more complicated, being a structure rather than just a property (and a structure that should be preserved by proximal maps).
As for Additivity itself, this presumably corresponds to something more general in the world of profunctors related to limits and colimits, but I haven't figured it out yet.
Symmetry probably doesn't fit into this picture very well, but who knows?
Generalized uniform structures
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