topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
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fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
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continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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. (See apartness space for a point–set version and apartness relation for a point–point version.)
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 $B^{\mathsf{c}}$ is the complement of $B$ in $X$, i.e. $X\setminus B$.)
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 $x \in A$ |
Transitivity | If for every $D \subseteq X$, either $A \;\delta\; D^{\mathsf{c}}$ or $D \;\delta\; B$, then $A \;\delta\; B$ | If $A \bowtie B$, then for some $D \subseteq X$, both $A \bowtie D^{\mathsf{c}}$ and $D \bowtie B$ | If $A \ll B$, then for some $D \subseteq X$, both $A \ll D$ and $D \ll B$ |
Symmetry (weak) | if $A \;\delta\; B$ then $B \;\delta\; A$ | if $A \bowtie B$ then $B \bowtie A$ | if $A \ll B$ then $B^{\mathsf{c}} \ll A^{\mathsf{c}}$ |
Symmetry (strong) | $A \;\delta\; B$ iff $B \;\delta\; A$ | $A \bowtie B$ iff $B \bowtie A$ | $A \ll B$ iff $B^{\mathsf{c}} \ll A^{\mathsf{c}}$ |
Local Decomposability | If $\{x\}\bowtie B$, then for some $C$ and $D$, $\{x\}\bowtie D$, $C\bowtie B$, and $C\cup D = X$. | If $\{x\}\ll B$, then for some $C$ and $D$, $\{x\}\ll C$, $D\ll B$, and $C^{\mathsf{c}}\cup D = X$. | |
Decomposability | If $A\bowtie B$, then for some $C$ and $D$, $A\bowtie D$, $C\bowtie B$, and $C\cup D = X$. | If $A\ll B$, then for some $C$ and $D$, $A\ll C$, $D\ll B$, and $C^{\mathsf{c}}\cup D = X$. | |
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 $A \ll C_i$ for all $i$, then $A \ll \bigcap_i C_i$ |
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; see below). 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. In particular, it allows us to assert Reflexivity only for singletons, although this is often not done (to avoid mentioning points).
In classical mathematics, the weak Symmetry axioms are equivalent to the strong versions, while Decomposability is equivalent to Transitivity (and Local Decomposability is of course a special case of Decomposability). In constructive mathematics, strong Symmetry for $\ll$ is unreasonably strong (as is binary Additivity for $\delta$), while Local Decomposability and Decomposability appear genuinely stronger than Transitivity (but not unreasonably so); see remarks below. (We don’t bother to state a version of (Local) Decomposability for $\delta$ because the other axioms for $\delta$ seem unreasonably strong constructively, while Decomposability is automatic in classical mathematics.)
A topogeny is a relation that satisfies both forms of Isotony and all four forms of Additivity. A topogenous relation or quasipreproximity is a topogeny that also satisfies Reflexivity; a quasiproximity is a quasipreproximity that also satisfies Reflexivity and Transitivity. A topogeny is symmetric if it satisfies Symmetry; a preproximity is a symmetric quasipreproximity, and a proximity is a symmetric quasiproximity. A topogeny is separated if it satisfies Separation. A topogeny is perfect if it satisfies left Perfection, coperfect if it satisfies right Perfection, and biperfect if it satisfies both; 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). Constructively, a proximity space satisfying Regularity may be called proximally regular.
A (quasi)-(pre)-proximity space is a set equipped with a (quasi)-(pre)-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 Transitivity (so ‘pre’ is the default for them). 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 quasipreproximity. The terminology for Perfection also comes from syntopogenous spaces.
Transitivity is sometimes called normality, due to its superficial similarity with the notion of a normal space. However, the underlying topology of a proximity space need not be normal in the topological sense. If regarded as a separation axiom, the Transitivity axiom is more analogous to complete regularity than to normality, and at least in classical mathematics every proximity space is completely regular — but this is not true in constructive mathematics, or for quasi-proximity spaces, so it is debatable whether it should be attributed to the Transitivity axiom. When comparing proximities to topologies via syntopogenies, Transitivity is instead analogous to the fact that the closure of a set is closed (or the interior of a set is open), which we require of all topologies; thus arguably we should require it of all (quasi-)proximities. If this is removed from the definition of topological space, then we have a pretopological space, which is the origin of the ‘pre’ terminology above.
If $X$ and $Y$ are (quasi)-(pre)-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.
As remarked above, the strong Symmetry axiom for $\ll$ is too strong in constructive mathematics. In fact, if it is satisfied by a nontrivial space $X$, then excluded middle follows: for any truth value $P$ such that $\neg\neg P$ is true, let $A=X$ and $B = \{ x\in X \mid P \}$; then $B^{\mathsf{c}} = \emptyset \ll \emptyset = A^{\mathsf{c}}$, so strong Symmetry gives $A\ll B$, hence $A\subseteq B$ and so $P$ holds.
On the other hand, while the Symmetry axioms for $\bowtie$ and $\delta$ may seem stronger than the allowable weak symmetry axiom for $\ll$, this is arguably an illusion. For given a proximal neighborhood space with $\ll$ (and weak Symmetry), if we define $A\bowtie B$ to mean $A\ll B^{\mathsf{c}}$, we obtain a proximal apartness space (with Symmetry). In fact, the Transitivity axiom for $\bowtie$ implies that if $A\bowtie B$ then $A\bowtie (B^{\mathsf{c}})^{\mathsf{c}}$, so that (with Symmetry) $\bowtie$ is completely determined by its restriction to subsets that are complements.
In this sense, a proximal neighborhood space actually contains more information than a proximal apartness space; it is unclear whether the latter gives rise to the former constructively. On the other side, we can of course define $A\;\delta \; B$ to mean $\neg(A\bowtie B)$, but with this definition we cannot even prove binary Additivity for $\delta$. In fact, binary Additivity for $\delta$ seems too strong: it is apparently not satisfied even by a metric space.
We can, however, show the following constructively.
(Quasi-)proximal apartness spaces satisfying Decomposability are equivalent to (quasi-)proximal neighborhood spaces satisfying Decomposability.
As above, in one direction we define $A\bowtie B$ to mean $A\ll B^{\mathsf{c}}$. In the other, we define $A\ll B$ to mean that there exists a $C$ with $A\bowtie C$ and $B\cup C = X$. Most of the axioms are easy. To prove right binary Additivity for $\ll$, given $A\ll B$ and $A\ll C$, we have $A\bowtie D$ and $A\bowtie E$ with $D\cup B = X$ and $E\cup C = X$, whence $A\bowtie D\cup E$ with $(D\cup E) \cup (B\cap C) = X$ by distributing $\cup$ over $\cap$.
To prove Decomposability (and hence Transitivity) for $\ll$, given $A\ll B$, we have $A\bowtie C$ with $B\cup C = X$, whence by Decomposability for $\bowtie$ we have $D,E$ with $A\bowtie D$, $E\bowtie C$, and $D\cup E = X$. By Decomposability again, we have $D_1,D_2,E_1,E_2$ with $A\bowtie D_1$, $D_2\bowtie D$, and $D_1\cup D_2 = X$, while $E\bowtie E_1$, $E_2\bowtie C$, and $E_1\cup E_2=X$. Then $A\ll D_2$ (because of $D_1$) and $E_2 \ll B$ (because of $C$), while $D\subseteq D_2^{\mathsf{c}}$ (since $D_2\bowtie D$) and $E\subseteq E_2$ (because $E\bowtie E_1$ and so $E\subseteq E_1^{\mathsf{c}} \subseteq E_2$) and so $X = D\cup E = D_2^{\mathsf{c}} \cup E_2$.
Lastly, if $\bowtie$ satisfies Symmetry and $A\ll B$, we have $A\bowtie C$ with $B\cup C = X$, whence by Decomposability for $\bowtie$ we have $D,E$ with $A\bowtie D$, $E\bowtie C$, and $D\cup E = X$. Then by Symmetry, we have $B^{\mathsf{c}} \subseteq C\bowtie E$ while $X = E\cup D \subseteq E\cup A^{\mathsf{c}}$, so $B^{\mathsf{c}} \ll A^{\mathsf{c}}$.
To show that these constructions are inverses, if we start with $\ll$ then by a round-trip we obtain $A\ll' B$ defined to mean that there exists $C$ with $A\ll C^{\mathsf{c}}$ and $B\cup C = X$. Now on one hand, $B\cup C = X$ implies $C^{\mathsf{c}}\subseteq B$, so $A\ll C^{\mathsf{c}} \subseteq B$. But on the other, if $A\ll B$ then Decomposability gives $C$ and $D$ with $A\ll C$, $D\ll B$, and $C^{\mathsf{c}}\cup D = X$, whence $A \ll (C^{\mathsf{c}})^{\mathsf{c}}$ and $X = D \cup C^{\mathsf{c}} \subseteq B \cup C^{\mathsf{c}}$.
On the other side, if we start with $\bowtie$ then by a round-trip we obtain $A\bowtie' B$ defined to mean that there exists $C$ with $A\bowtie C$ and $C \cup B^{\mathsf{c}}= X$. Now on one hand, if $C \cup B^{\mathsf{c}}= X$ then $B \subseteq C$, so $A\bowtie C$ implies $A\bowtie B$. But on the other, if $A\bowtie B$ then Decomposability gives $C$ and $D$ with $A\bowtie C$ and $D\bowtie B$ and $C\cup D = X$, whence $D\subseteq B^{\mathsf{c}}$ and so $X = C\cup D \subseteq C \cup B^{\mathsf{c}}$.
In other words, the two constructively reasonable ways to define a (quasi-)proximity space give the same result in the Decomposable case.
On the other hand, the Decomposability axiom is very strong, constructively; as noted below, the standard way to make a proximity space from a uniform space does not satisfy it. Thus constructive treatments of proximity such as Bridges-Vita have usually not included Decomposability, instead using the weaker Transitivity axiom (called the Efremovic property in Bridges-Vita).
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 Transitivity, $\leq$ 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 $SymPreOrd$ of symmetric prosets (sets equipped with equivalence relations) as a reflexive subcatgory of $Prox$.
In constructive mathematics, a quasi-proximal nearness space (despite its general unreasonableness) or neighborhood space gives rise to a preordered set as above, while a proximal apartness space yields an inequality relation that is an apartness relation if we have Local Decomposability.
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 is 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 $Cl(A) \subseteq Int(B)$.
The construction of a topology works just as well for a quasi-proximity, but the result is no longer necessarily completely regular. In fact, every topology can be induced by some quasi-proximity: define $A \ll B$ if $A \subseteq int(B)$. Moreover, in this way topological spaces can be identified with (left) perfect quasi-proximity spaces.
In constructive mathematics, the three now-inequivalent kinds of (quasi-)proximity space (nearness, apartness, and neighborhood) yield three inequivalent notions of topological space: closure operators, point-set apartness spaces, and topological spaces (equivalently neighborhood spaces). Note in particular that the Transitivity axiom ensures that (in the three cases) the closure of a set is closed, that the final axiom of a point-set apartness space holds (if $x\bowtie A$ then $x\bowtie \{ y \mid \neg(y\bowtie A) \}$), and that the interior of a set is open. Hence, we can identify left perfect quasi-proximal apartness or neighborhood spaces with point-set apartness spaces and topological spaces respectively, by defining $A \ll B$ if $A \subseteq int(B)$ as above in the second case, and $A\bowtie B$ if $x\in A \Rightarrow x\bowtie B$ in the first.
If a quasi-proximal apartness or neighborhood space satisfies Local Decomposability, then its underlying point-set apartness space or topological space is locally decomposable (hence the name for the former axiom). Conversely, the canonical quasi-proximial apartness or neighborhood space obtained from a point-set apartness space or topological space satisfies Local Decomposability if the original space was locally decomposable.
Unlike in the classical case, the topology underlying a proximity space (which, recall, includes Symmetry) apparently need not be completely regular or even regular. However, we can say:
Let $X$ be a proximal apartness or neighborhood space (with Symmetry). Then:
Suppose first that $X$ is a Locally Decomposable proximal neighborhood space and $U$ is an open set containing $x$, which means $\{x\} \ll U$. By Local Decomposability, we have sets $C$ and $D$ such that $\{x\} \ll C$ and $D\ll U$ and $X = D \cup C^{\mathsf{c}}$. Transitivity then gives us a set $B$ with $\{x\} \ll B \ll C$, and Symmetry then tells us that $C^{\mathsf{c}} \ll B^{\mathsf{c}}$. Let $V = int(B^{\mathsf{c}})$ and $W = \int(B)$; then $V\cap W = \emptyset$, $W$ is an open neighborhood of $x$, and $X = D \cup C^{\mathsf{c}} \subseteq U \cup V$. Thus, $X$ is regular.
Now suppose $X$ is a Decomposable proximal neighborhood space and we have countable choice. Then given any opens $A,B$ with $A\ll B$, essentially the same proof as above shows that $A$ is “well-inside” $B$. Thus, applying Transitivity repeatedly (making countably many choices) we can obtain a “scale” of interpolating well-inside open subsets indexed by (dyadic) rational numbers, and thereby a function $f:X\to \mathbb{R}$ separating $A$ and $B$, as in the classical proof of Urysohn’s lemma. (With only Transitivity instead of Decomposability, we would obtain only a function valued in the lower or upper real numbers, rather than the located real numbers.)
The arguments for proximal apartness spaces are essentially the same.
Conversely, if $X$ is a completely regular topological space, then the above definition that $A \bowtie B$ iff they are separated by a continuous function $f: X\to [0,1]$ is also constructively valid, and produces a proximity space satisfying Symmetry and Decomposability.
If $Y$ is 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 defines a biperfect proximity structure on $Y$. In terms of the other relations, this means $A\bowtie B$ iff $V \cap (A \times B) = \emptyset$ for some entourage $V$, and $A\ll B$ iff $V[A] \subseteq B$ for some entourage $V$, where $V[A] = \{ y \mid \exists x\in A, (x,y)\in V \}$.
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 between given uniform spaces 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.
The relation between quasi-uniformities and quasi-proximities is similar.
In constructive mathematics, all three of the relations $\delta,\bowtie,\ll$ may be defined as above from a uniformity. All the axioms of all three kinds of proximity are then provable, except Binary Additivity for $\delta$, Strong Symmetry for $\ll$, and (Local) Decomposability. Local Decomposability follows from uniform regularity for the uniform space, but Decomposability seems unobtainable in this way; it seems to assert locatedness of all subsets.
Moreover, constructively not every $\bowtie$- or $\ll$-proximity is induced by a uniformity in this way. Consider, on an inhabited set $X$, the relation $A\bowtie B$ defined to mean $A = \emptyset \vee B=\emptyset$. This is a $\bowtie$-proximity, but if it is induced by a uniformity, then weak excluded middle holds. To see this, given a proposition $P$, fix $\xi\in X$ and let $A = \{ x \mid x=\xi \wedge P\}$ and $B = \{x \mid x=\xi \wedge \neg P\}$. Then $A\times B = \emptyset$, hence $A\bowtie B$ in any proximity defined from a uniformity; but to say $A = \emptyset \vee B=\emptyset$ is weak excluded middle for $P$. (This example is from Bridges and Vita.)
However, there is a different way to obtain a proximity underlying a uniformity: define $A\bowtie B$ if there is a uniformly continuous function $f: X\to [0,1]$ such that $f(x) = 0$ for $x \in A$ and $f(x) = 1$ for $x \in B$. Classically, this is equivalent to the usual definition, by the usual proof that any uniform space is completely regular. Constructively, it (and a similar definition for $\ll$) yields a proximity space satisfying Decomposability, which has the same underlying topology if our original uniform space was already completely regular. It seems possible that at least with countable choice, we could show that every Decomposable proximity space arises from a uniform space in this way.
A proximity space can be identified with a syntopogenous space which is both simple and symmetric, and similarly a quasi-proximity space can be identified with one that is just simple. The relations with topological spaces and uniform spaces can then be seen as coreflections inside the category of syntopogenous spaces; 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
Note that conditions (1) and (2), together with Isotony for $\delta$, imply that any cluster $\sigma$ is upwards-closed. Together with nullary Additivity for $\delta$, they also imply the nullary version of (3): $\emptyset \notin\sigma$. However, a cluster is not generally a filter: it is not closed under binary intersections. This is most obvious when we consider how $X$ embeds in its compactification, taking each point $x$ to the “principal cluster” $\sigma_x = \{ A \mid \{x\}\;\delta\; A \}$, i.e. the collection of sets whose closure contains $x$, which is certainly not generally closed under intersections. (If it were a filter, of course, condition (3) would say that it is prime.)
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 Transitivity 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 Transitivity 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 Transitivity 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, Peter Schuster, and Luminita Vita, Apartness, topology, and uniformity: a constructive view, pdf
Douglas Bridges and Luminita Vita, Apartness and Uniformity: a constructive development
S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge University Press 1970
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