nLab
syntopogenous space

Idea

A syntopogenous space is a common generalization of topological spaces, proximity spaces, and uniform spaces. The category of syntopogenous spaces includes $\mathrm{Top}$, $\mathrm{Prox}$, and $\mathrm{Unif}$ as full subcategories whose intersection is fairly trivial.

Definitions

Topogenous relations

A binary relation $\delta$ on the power set $P(X)$ of a set $X$ is called topogenous if it satisfies:

1. nontriviality: if $A\cap B$ is inhabited, then $A\delta B$.

2. binary additivity: $A\delta (B\cup C)$ if and only if either $A\delta B$ or $A\delta C$.

3. nullary additivity: it is never true that $A\delta \varnothing$ or $\varnothing \delta A$ for any $A$.

Note that the “if” direction of binary additivity is equivalent to isotony: if $A\subseteq C$ and $B\subseteq D$, then $A\delta B$ implies $C\delta D$.

The set of topogenous relations on $X$, ordered by containment, is a complete lattice:

• Its least element is the discrete topogenous relation, defined by $A\delta B$ if and only if $A\cap B$ is inhabited.
• Its greatest element is the codiscrete topogenous relation, defined by $A\delta B$ if and only if both $A$ and $B$ are inhabited.
• The union of any inhabited set of topogenous relations is topogenous, hence is a join. The same is true for directed intersections.
• The meet of a non-directed set $𝒟$ of topogenous relations is not their set-theoretic intersection, but it can be described explicitly: we have $A(\bigwedge 𝒟)B$ if and only if whenever $A={\bigcup }_{i=1}^n{A}_i$ and $B={\bigcup }_{j=1}^m{B}_j$, there exist $i$ and $j$ such that $A_i\delta B_j$ for all $\delta \in 𝒟$.

The opposite relation of a topogenous relation is again topogenous. A topogenous relation is called symmetric if it is equal to its opposite, i.e. if $A\delta B$ if and only if $B\delta A$.

A topogenous relation is called perfect if $A\delta B$ implies there exists an $x\in A$ with $\{x\}\delta B$. It is called biperfect if both it and its opposite are perfect. Of course, a symmetric perfect topogenous relation is automatically biperfect.

Syntopogenous spaces

A syntopogeny (or syntopogenous structure) on a set $X$ is a filter $𝒪$ of topogenous relations such that

• For any $\delta \in 𝒪$, there exists a $\delta \prime \in 𝒪$ such that if $A,B\subseteq X$ have the property that whenever $C\cup D=X$, either $A\delta \prime C$ or $B\delta \prime D$, then $A\delta B$.

A basis for a syntopogeny on $X$ is a filterbase in the complete lattice of topogenous structures, such that the filter it generates is a syntopogeny. When $X$ is equipped with a syntopogeny, it is called a syntopogenous space.

A syntopogeny is called symmetric, perfect, or biperfect if it admits a basis consisting of symmetric, perfect, or biperfect topogenous relations, respectively. It is called simple if it admits a basis that is a singleton.

Syntopogenous functions

If $\delta$ is a topogenous relation on $Y$ and $f:X\to Y$ is a function, then we have a topogenous relation $f^{*}\delta$ on $X$ defined by $A(f^{*}\delta )B$ iff $f(A)\delta f(B)$. That is, $f^{*}\Delta =({\exists }_f\times {\exists }_f{)}^{-1}(\delta )$, where ${\exists }_f:P(X)\to P(Y)$ is the left adjoint of $f^{-1}:P(Y)\to P(X)$.

Now if $(X_1,{𝒪}_1)$ and $(X_2,{𝒪}_2)$ are syntopogenous spaces, a function $f:X_1\to X_2$ is called syntopogenous, or syntopologically continuous, if for any $\delta \in {𝒪}_2$, we have $f^{*}\delta \in {𝒪}_1$. This defines the category $\mathrm{STpg}$.

If $(Y,𝒪)$ is a syntopogenous space, then the collection $\{f^{*}\delta \mid \delta \in 𝒪\}$ is a basis for a syntopogeny on $X$, which is the initial structure induced on $X$ by $f$. The operation of taking initial structures, as a map from syntopogenies on $Y$ to syntopogenies on $X$, preserves opposites, simplicity, meets, symmetry, and perfectness.

More generally, if $(Y_i,{𝒪}_i)$ is a family of syntopogenous spaces and $f_i:X\to Y_i$ are functions, then the meet of the initial structures induced by all the $f_i$ is the initial structure induced by them jointly. Thus, $\mathrm{STpg}\to \mathrm{Set}$ is a topological concrete category.

Relation to other topological structures

Topological spaces

If $X$ is a topological space, we define $A\delta B$ to hold if $A\cap \overline{B}$ is inhabited, where $\overline{B}$ denotes the closure of $B$. This is a basis for a simple perfect syntopogeny.

Conversely, given a simple perfect syntopogeny, with singleton basis $\{\delta \}$, we define $\overline{B}=\{x\mid \{x\}\delta B\}$; then this is a Kuratowski closure operator and hence defines a topology.

These constructions define an equivalence of categories between Top and the full subcategory of $\mathrm{STpg}$ on the simple, perfect, syntopogenous spaces.

Proximity spaces

A simple symmetric syntopogeny is easily seen to be precisely a proximity. In this way we have an equivalence of categories between $\mathrm{Prox}$ and the full subcategory of $\mathrm{STpg}$ on the simple, symmetric, syntopogenous spaces.

More generally, an arbitrary simple syntopogeny can be identified with a “quasi-proximity”: a non-symmetric relation satisfying all the other axioms of a proximity (suitably rephrased for the non-symmetric case).

Uniform spaces

If $\delta$ is a biperfect syntopogenous relation, then we have $A\delta B$ if and only if there exist $x\in A$ and $y\in B$ with $\{x\}\delta \{y\}$. Therefore, $\delta$ is completely determined by a binary relation $U\subseteq X\times X$ on $X$, which contains the diagonal ${\Delta }_X$. Conversely, any binary relation on $X$ containing the diagonal defines a biperfect syntopogenous relation.

It follows that biperfect syntopogenies are equivalent to quasi-uniformities, which are like uniformities but lack the symmetry axiom. We have an equivalence of categories between $\mathrm{QUnif}$ and the full subcategory of $\mathrm{STpg}$ on the biperfect syntopogenous spaces, which easily restricts to an equivalence between $\mathrm{Unif}$ and the category of symmetric, (bi)perfect topogenous spaces.

Preorders and setoids

A syntopogeny which is both simple and biperfect is determined uniquely by a single relation on $X$ which must be both reflexive and transitive, i.e. a preorder. Thus, the intersection $\mathrm{Top}\cap \mathrm{QUnif}$ inside $\mathrm{STpg}$ is equivalent to $\mathrm{Preord}$.

Of course, it follows that a simple, symmetric, (bi)perfect syntopogeny is determined uniquely by a relation on $X$ that is reflexive, transitive, and also symmetric – i.e. an equivalence relation. Thus, the intersections $\mathrm{Top}\cap \mathrm{Unif}$, $\mathrm{Top}\cap \mathrm{Prox}$, and $\mathrm{Prox}\cap (Q)\mathrm{Unif}$ inside $\mathrm{STpg}$ are all equivalent to the category $\mathrm{Setoid}$ of setoids (sets equipped with an equivalence relation).

Some coreflections

In the preorder of topogenous relations on any set $X$, the following sub-preorders are coreflective:

• The symmetric elements. The symmetric coreflection of $\delta$ is the meet ${\delta }^s≔\delta \wedge {\delta }^{\mathrm{op}}$.
• The perfect elements. The perfect coreflection of $\delta$ is defined by $A{\delta }^{p}B$ iff there exists $x\in A$ with $\{x\}\delta B$.
• The biperfect elements. The byperfect coreflection of $\delta$ is defined by $A{\delta }^{b}B$ iff there exist $x\in A$ and $y\in B$ with $\{x\}\delta \{y\}$.

It follows that in the preorder of syntopogenous structures on $X$, the symmetric, perfect, and biperfect elements are also reflective; the coreflections are obtained by applying the above one to each topogenous relation in turn. Moreover, the simple syntopogenous structures on $X$ are also coreflective; the coreflection just takes the intersection of all relations belonging to the filter (this is a directed intersection, hence automatically again a topogenous relation).

Finally, for any function $f:X\to Y$, the preimage function $f^{*}$, mapping syntopogenous structures on $Y$ to those on $X$, preserves all of these coreflections. Therefore, the full subcategories of

• simple,
• symmetric,
• perfect, and
• biperfect

syntopogenous spaces are all coreflective in $\mathrm{STpg}$, with coreflections written $(-{)}^t$, $(-{)}^s$, $(-{)}^p$, and $(-{)}^b$ respectively.

In general, of course, coreflections into distinct subcategories do not commute or even preserve each other’s subcategories. However, by construction, we see that the coreflections $(-{)}^s$, $(-{)}^p$, and $(-{)}^b$ all preserve simplicity. Therefore, the full subcategories of

• simple symmetric (i.e. proximity),
• simple perfect (i.e. topological), and
• simple biperfect (i.e. preorders)

syntopogenous spaces are all coreflective in $\mathrm{STpg}$, with coreflections $(-{)}^{ts}$, $(-{)}^{tp}$, and $(-{)}^{tb}$ respectively. Finally, it is evident by construction that $(-{)}^b$ preserves symmetry, so the full subcategories of

• symmetric biperfect, and
• simple symmetric biperfect (i.e. setoids)

syntopogenous spaces are also both coreflective in $\mathrm{STpg}$, with coreflections $(-{)}^{sb}$ and $(-{)}^{tsb}$ respectively.

It is straightforward to verify the following.

1. When applied to a (quasi-)proximity space or a (quasi-)uniform space, the coreflection $(-{)}^{tp}$ into topological spaces computes the underlying topology of these structures, as usually defined.

2. When applied to a uniform space, the coreflection $(-{)}^{ts}$ computes its underlying proximity, as usually defined. The same is true in the non-symmetric case for quasi-uniformities and quasi-proximities.

3. When applied to any syntopogenous space, the coreflection $(-{)}^{tb}$ computes the specialization order of its underlying topology (i.e. its image under $(-{)}^{tp}$). In particular, this is the case for topological spaces, proximity spaces, and uniform spaces.

References

• Ákos Császár, Foundations of General Topology, 1963

Császár speaks of topogenous orders $\ll$ rather than our topogenous relations; there is a one-to-one correspondence between them, defined by $A\ll B$ if and only if it is not the case that $A\delta (X\setminus B)$. One could equivalently axiomatize the negation of $\delta$ itself, and so on. Of course, in constructive mathematics these will no longer be equivalent, and one must make a suitable choice.

Note that the containment relation on topogenous orders is reversed from topogenous relations, so that instead of filters we have ideals. Császár also defines a syntopogenous structure to be what we have called a basis for one. As usual, this is convenient for concreteness (especially in the simple case), but has the disadvantage that distinct structures can nevertheless be isomorphic via an identity function, i.e. the forgetful functor to $\mathrm{Set}$ is not amnestic. On this page, we have followed the traditional practice for other topological structures in choosing to make this functor amnestic.