CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A syntopogenous space is a common generalization of topological spaces, proximity spaces, and uniform spaces. The category of syntopogenous spaces includes $Top$, $Prox$, and $Unif$ as full subcategories whose intersection is fairly trivial.
A binary relation $\delta$ on the power set $P(X)$ of a set $X$ is called topogenous if it satisfies:
Nontriviality or reflexivity: if $A \cap B$ is inhabited, then $A \;\delta\; B$.
Binary additivity: $A \;\delta\; (B \cup C)$ if and only if $A \;\delta\; B$ or $A \;\delta\; C$; and $(A \cup B) \;\delta\; C$ if and only if $A \;\delta\; C$ or $B \;\delta\; C$.
Nullary additivity: it is never true that $A \;\delta\; \emptyset$ or $\emptyset \;\delta\; A$ for any $A$.
Note that the “if” directions of binary additivity are equivalent to isotony: if $A \supseteq B \;\delta\; C \subseteq D$ implies $A \;\delta\; D$. We might specify these separately and call only the reverse direction of binary additivity ‘binary additivity’.
A relation satisfying merely (2) and (3) is called a topogeny (between $P(X)$ and itself) at topogeny; it is slightly easier to work with the lattice of topogenies than the lattice of topogenous relations, but syntopogenous spaces are built only out of those topogenies that satisfy (1).
As is shown at topogeny, a topogenous relation can also be regarded as equivalent to a reflexive relation from $\beta X$ to $\beta X$ (see ultrafilter monad) in the pretopos of compact Hausdorff spaces. In more concrete terms: each topogeny $\delta$ is a union of “basic topogenies” which are those of the form $\mathcal{U} \times \mathcal{V} \subset P X \times P X$ where $\mathcal{U}, \mathcal{V}$ are ultrafilters, and the set of all pairs $(\mathcal{U}, \mathcal{V})$ whose product is contained in $\delta$ forms a closed subspace of $\beta X \times \beta X$.
It is also sometimes convenient to work with topogenous relations (or topogenies) that are described by the negation of $\delta$ (written $\bowtie$) or by the topogenous order $\ll$ (which is transitive by (1) and isotony) given by $A \ll B$ iff $A \bowtie (X \setminus B)$. In constructive mathematics, these choices potentially make a difference. See proximity space for complete definitions.
The set of topogenous relations on $X$, ordered by containment, is a complete lattice, as is the set of topogenies. (If we are working with $\bowtie$ or $\ll$ instead of with $\delta$, then the order relation in these lattices is reversed).
From binary meets (just above), directed meets (above that) and nullary meets (the greatest element), we get all meets (even constructively). But the meet of an arbitrary set $\mathcal{D}$ of topogenous relations (or topogenies) can still be easily described explicitly: we have $A \;(\bigwedge\mathcal{D})\;B$ 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 $\delta\in\mathcal{D}$.
The opposite relation of a topogenous relation is again topogenous. A topogenous relation (or topogeny) 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.
A syntopogeny (or syntopogenous structure) on a set $X$ is a filter $\mathcal{O}$ of topogenous relations (or an ideal if we are working with $\bowtie$ or $\ll$) such that
If we have been working with the lattice of topogenies rather than topogenous relations, then we must explicitly state here that every topogeny in $\mathcal{O}$ satisfies (1). This requirement actually joins with the requirement above to form an unbiased definition that is nicely expressed in terms of topogenous orders $\ll$ (and tacitly using isotony to show the equivalence):
A basis for a syntopogeny on $X$ is a filterbase in the complete lattice of topogenous relations (or of topogenies), 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 (which must then be a quasiproximity, even a proximity in the symmetric case).
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 $(X_1,\mathcal{O}_1)$ and $(X_2,\mathcal{O}_2)$ are syntopogenous spaces, a function $f:X_1\to X_2$ is called syntopogenous, or syntopologically continuous, if for any $\delta\in\mathcal{O}_2$, we have $f^*\delta\in\mathcal{O}_1$. This defines the category $STpg$.
If $(Y,\mathcal{O})$ is a syntopogenous space, then the collection $\{ f^*\delta | \delta\in\mathcal{O}\}$ 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 $(Y_i,\mathcal{O}_i)$ 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, $STpg\to Set$ is a topological concrete category.
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 $\overline{B} = \{ x | \{x\}\;\delta\; B \}$; 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 $STpg$ on the simple, perfect, syntopogenous spaces.
A simple symmetric syntopogeny is easily seen to be precisely a proximity. In this way we have an equivalence of categories between $Prox$ and the full subcategory of $STpg$ on the simple, symmetric, syntopogenous spaces.
More generally, an arbitrary simple syntopogeny can be identified with a quasiproximity, and we have an equivalence of categories between $QPrxo$ and the subcategory of simple syntopogenous spaces.
If $\delta$ is a biperfect topogenous 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 topogenous relation. (For a biperfect topogeny, remove the requirement that $U$ contain the diagonal.)
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 $QUnif$ and the full subcategory of $STpg$ on the biperfect syntopogenous spaces, which easily restricts to an equivalence between $Unif$ and the category of symmetric, (bi)perfect topogenous spaces.
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 $Top \cap QUnif$ inside $STpg$ is equivalent to $Preord$.
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 $Top \cap Unif$, $Top \cap Prox$, and $Prox\cap (Q)Unif$ inside $STpg$ are all equivalent to the category $Setoid$ of setoids (sets equipped with an equivalence relation).
In the preorder of topogenous relations on any set $X$, the following sub-preorders are coreflective:
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
syntopogenous spaces are all coreflective in $STpg$, 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
syntopogenous spaces are all coreflective in $STpg$, with coreflections $(-)^{t s}$, $(-)^{t p}$, and $(-)^{t b}$ respectively. Finally, it is evident by construction that $(-)^b$ preserves symmetry, so the full subcategories of
syntopogenous spaces are also both coreflective in $STpg$, with coreflections $(-)^{s b}$ and $(-)^{t s b}$ respectively.
It is straightforward to verify the following.
When applied to a (quasi)-proximity space or a (quasi)-uniform space, the coreflection $(-)^{t p}$ into topological spaces computes the underlying topology of these structures, as usually defined.
When applied to a uniform space, the coreflection $(-)^{t s}$ computes its underlying proximity, as usually defined. The same is true in the non-symmetric case for quasi-uniformities and quasi-proximities.
When applied to any syntopogenous space, the coreflection $(-)^{t b}$ computes the specialization order of its underlying topology (i.e. its image under $(-)^{t p}$). In particular, this is the case for topological spaces, proximity spaces, and uniform spaces.
Generalized uniform structures
Császár uses topogenous orders $\ll$. 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 $Set$ is not amnestic. On this page, we have followed the traditional practice for other topological structures in choosing to make this functor amnestic.