nLab
syntopogenous space

Syntopogenous spaces

Idea

A syntopogenous space is a common generalization of topological spaces, proximity spaces, and uniform spaces. The category of syntopogenous spaces includes TopTop, ProxProx, and UnifUnif as full subcategories whose intersection is fairly trivial.

Definitions

Topogenous relations

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

  1. Nontriviality or reflexivity: if ABA \cap B is inhabited, then AδBA \;\delta\; B.

  2. Binary additivity: Aδ(BC)A \;\delta\; (B \cup C) if and only if AδBA \;\delta\; B or AδCA \;\delta\; C; and (AB)δC(A \cup B) \;\delta\; C if and only if AδCA \;\delta\; C or BδCB \;\delta\; C.

  3. Nullary additivity: it is never true that AδA \;\delta\; \emptyset or δA\emptyset \;\delta\; A for any AA.

Note that the “if” directions of binary additivity are equivalent to isotony: if ABδCDA \supseteq B \;\delta\; C \subseteq D implies AδDA \;\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)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).

It is also sometimes convenient to work with topogenous relations (or topogenies) that are described by the negation of δ\delta (written \bowtie) or by the topogenous order \ll (which is transitive by (1) and isotony) given by ABA \ll B iff A(XB)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 XX, ordered by containment, is a complete lattice, as is the set of topogenies. (If we are working with \bowtie or \ll instead of with δ\delta, then the order relation in these lattices is reversed).

  • The least element is the discrete topogenous relation, defined by AδBA \;\delta\; B if and only if ABA \cap B is inhabited. (In the set of topogenies, the least element is the trivial topogeny, in which AδBA \;\delta\; B never holds.)
  • The greatest element is the codiscrete topogenous relation (also the cotrivial topogeny), defined by AδBA \;\delta\; B if and only if both AA and BB are inhabited.
  • The union of any inhabited set of topogenous relations is topogenous, hence is a join. To treat inhabited joins and the nullary join (the least element) uniformly, we can take the union of the topogenous relations and the discrete topogenous relation. (With topogenies, we can just take the union, period.)
  • The intersection of a directed set of topogenous relations (or topogenies) is again a topogenous relation (or topogeny) and so is a meet.
  • In general, the intersection of two topogenous relations is not topogenous (or even a topogeny); however, they still have a meet: AδBA \;\delta\; B (where δ\delta is the meet of δ 1\delta_1 and δ 2\delta_2) iff, whenever A= i=1 nA iA = \bigcup_{i=1}^n A_i and B= j=1 mB jB = \bigcup_{j=1}^m B_j, there exist ii and jj such that A iδ 1B jA_i \;\delta_1\; B_j and A iδ 2B jA_i \;\delta_2\; B_j.

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 𝒟\mathcal{D} of topogenous relations (or topogenies) can still be easily described explicitly: we have A(𝒟)BA \;(\bigwedge\mathcal{D})\;B if and only if whenever A= i=1 nA iA = \bigcup_{i=1}^n A_i and B= j=1 mB jB = \bigcup_{j=1}^m B_j, there exist ii and jj such that A iδB jA_i \;\delta\; B_j for all δ𝒟\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 AδBA \;\delta\; B if and only if BδAB \;\delta\; A.

A topogenous relation is called perfect if AδBA \;\delta\; B implies there exists an xAx \in A with {x}δB\{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 XX is a filter 𝒪\mathcal{O} of topogenous relations (or an ideal if we are working with \bowtie or \ll) such that

  • For any δ𝒪\delta\in \mathcal{O}, there exists a δ𝒪\delta'\in\mathcal{O} such that if A,BXA,B\subseteq X have the property that whenever CD=XC\cup D = X, either AδCA\;\delta'\; C or BδDB\;\delta'\; D, then AδBA\;\delta\; B.

If we have been working with the lattice of topogenies rather than topogenous relations, then we must explicitly state here that every topogeny in 𝒪\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):

  • For each natural number nn and each \ll in the syntopogeny 𝒪\mathcal{O}, there is an \ll' in 𝒪\mathcal{O} such that, whenever ABA \ll B, there is a list C 0,,C nC_0, \ldots, C_n such that AC 0C 1C nBA \subseteq C_0 \ll C_1 \cdots \ll C_n \subseteq B.

A basis for a syntopogeny on XX is a filterbase in the complete lattice of topogenous relations (or of topogenies), such that the filter it generates is a syntopogeny. When XX 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).

Syntopogenous functions

If δ\delta is a topogenous relation on YY and f:XYf:X\to Y is a function, then we have a topogenous relation f *δf^*\delta on XX defined by A(f *δ)BA\;(f^*\delta)\;B iff f(A)δf(B)f(A) \;\delta\; f(B). That is, f *Δ=( f× f) 1(δ)f^*\Delta = (\exists_f \times \exists_f)^{-1}(\delta), where f:P(X)P(Y)\exists_f : P(X) \to P(Y) is the left adjoint of f 1:P(Y)P(X)f^{-1}:P(Y) \to P(X).

Now if (X 1,𝒪 1)(X_1,\mathcal{O}_1) and (X 2,𝒪 2)(X_2,\mathcal{O}_2) are syntopogenous spaces, a function f:X 1X 2f:X_1\to X_2 is called syntopogenous, or syntopologically continuous, if for any δ𝒪 2\delta\in\mathcal{O}_2, we have f *δ𝒪 1f^*\delta\in\mathcal{O}_1. This defines the category STpgSTpg.

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

More generally, if (Y i,𝒪 i)(Y_i,\mathcal{O}_i) is a family of syntopogenous spaces and f i:XY if_i:X\to Y_i are functions, then the meet of the initial structures induced by all the f if_i is the initial structure induced by them jointly. Thus, STpgSetSTpg\to Set is a topological concrete category.

Relation to other topological structures

Topological spaces

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

Conversely, given a simple perfect syntopogeny, with singleton basis {δ}\{\delta\}, we define B¯={x|{x}δB}\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 STpgSTpg 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 ProxProx and the full subcategory of STpgSTpg 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 QPrxoQPrxo and the subcategory of simple syntopogenous spaces.

Uniform spaces

If δ\delta is a biperfect topogenous relation, then we have AδBA\;\delta\;B if and only if there exist xAx\in A and yBy\in B with {x}δ{y}\{x\}\;\delta\;\{y\}. Therefore, δ\delta is completely determined by a binary relation UX×XU\subseteq X\times X on XX, which contains the diagonal Δ X\Delta_X. Conversely, any binary relation on XX containing the diagonal defines a biperfect topogenous relation. (For a biperfect topogeny, remove the requirement that UU 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 QUnifQUnif and the full subcategory of STpgSTpg on the biperfect syntopogenous spaces, which easily restricts to an equivalence between UnifUnif 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 XX which must be both reflexive and transitive, i.e. a preorder. Thus, the intersection TopQUnifTop \cap QUnif inside STpgSTpg is equivalent to PreordPreord.

Of course, it follows that a simple, symmetric, (bi)perfect syntopogeny is determined uniquely by a relation on XX that is reflexive, transitive, and also symmetric – i.e. an equivalence relation. Thus, the intersections TopUnifTop \cap Unif, TopProxTop \cap Prox, and Prox(Q)UnifProx\cap (Q)Unif inside STpgSTpg are all equivalent to the category SetoidSetoid of setoids (sets equipped with an equivalence relation).

Some coreflections

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

  • The symmetric elements. The symmetric coreflection of δ\delta is the meet δ sδδ op\delta^s \coloneqq \delta \wedge \delta^{op}.
  • The perfect elements. The perfect coreflection of δ\delta is defined by Aδ pBA\;\delta^p\;B iff there exists xAx\in A with {x}δB\{x\}\;\delta\;B.
  • The biperfect elements. The byperfect coreflection of δ\delta is defined by Aδ bBA\;\delta^b\;B iff there exist xAx\in A and yBy\in B with {x}δ{y}\{x\}\;\delta\;\{y\}.

It follows that in the preorder of syntopogenous structures on XX, 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 XX 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:XYf:X\to Y, the preimage function f *f^*, mapping syntopogenous structures on YY to those on XX, preserves all of these coreflections. Therefore, the full subcategories of

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

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

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

syntopogenous spaces are also both coreflective in STpgSTpg, with coreflections () sb(-)^{s b} and () tsb(-)^{t s b} respectively.

It is straightforward to verify the following.

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

  2. When applied to a uniform space, the coreflection () ts(-)^{t s} 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(-)^{t b} computes the specialization order of its underlying topology (i.e. its image under () tp(-)^{t p}). In particular, this is the case for topological spaces, proximity spaces, and uniform spaces.

Generalized uniform structures

proarrowmonadpro-monadsymmetric versions
binary relationpreorderquasiuniformitysymmetric relationequivalence relationuniformity
binary function to [0,)[0,\infty)quasipseudometricquasiprometricsymmetric binary functionpseudometricprometric
topogenyquasiproximitysyntopogenysymmetric topogenyproximitysymmetric syntopogeny

References

  • Ákos Császár, Foundations of General Topology, 1963

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 SetSet is not amnestic. On this page, we have followed the traditional practice for other topological structures in choosing to make this functor amnestic.

Revised on July 27, 2014 09:17:21 by Toby Bartels (98.23.133.202)