@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 1, 2003 pp. 91–97 Functorial approach structures G. C. L. Brümmer and M. Sioen Abstract. We show that there exists at least a proper class of functorial approach structures, i.e., right inverses to the forgetful func- tor T : AP → Top (where AP denotes the topological construct of approach spaces and contractions as introduced by R. Lowen). There is however a great difference in nature of these functorial approach structures when compared to the quasi-uniform paradigm which has been extensively studied by the first author: whereas it is well-known from [2] that a large class of epireflective subcategories of Top0 can be ‘parametrized’ using the interaction of functorial quasi-uniformities with the quasi-uniform bicompletion, we show that using functorial ap- proach structures together with the approach bicompletion developed in [10], only Top0 itself can be retrieved in this way. 2000 AMS Classification: 18B30, 18B99, 54B30, 54E15, 54E99. Keywords: Approach space, (approach) bicompleteness, epireflective sub- category, functorial approach structure, spanning, topological space. 1. Introduction and Preliminaries In [2, 3, 4, 5, 7, 8, 9] so-called functorial (quasi-)uniformities were extensively studied. A functorial quasi-uniformity, is a functor F : Top → QU (where Top, resp. QU, stands for the topological construct of topological spaces and continuous maps, resp. of quasi-uniform spaces and uniformly continuous maps) which is a section for the usual forgetful functor Tqu : QU → Top, i.e. such that TquF = 1Top. First of all, let us recall from [3, 9] that there is a one-to-one correspondence between functorial quasi-uniformities in the above sense, and functorial quasi-uniformities F : Top0 → QU0 in the T0 case. We refer to [1] as our blanket reference for categorical material and to [11] for all information about quasi-uniformities. Let us only mention that for the order in the fibres of a topological construct A, we take the opposite convention to the one taken in [1]: if A,B are two objects on the same underlying set, we call A finer than B (or B coarser than A), and write A ≥ B, iff the identity map on the underlying set becomes a morphism A → B. Then all the fibres 92 G. C. L. Brümmer and M. Sioen become complete lattices. One of the most important results about functorial quasi-uniformities, is their interplay with the quasi-uniform bicompletion (cf. [2]). It was shown by the first named author that functorial quasi-uniformities can e.g. be used to classify epireflective subcategories of Top0, in the sense that for every (full) epireflective subcategory E of Top0 with |Sob| ⊆ |E| ⊆ |TopBicompl0| (where Sob stands for the subcategory of Top0 formed by all sober objects and TopBicompl0 for the one formed by all topologically bicomplete T0-spaces (i.e. those T0 topological spaces admitting a bicomplete quasi-uniformity)), there exists a functorial quasi-uniformity F : Top0 → QU0 such that (1.1) E = {X ∈ Top0 | FX bicomplete}. We refer to [2] for a survey on this topic. In [13], the topological construct AP of approach spaces and contractions was introduced by R. Lowen as a quantified supercategory of Top and it has been proved since then that approach spaces are an interesting framework for quantitative topology (see e.g. [14, 15, 16, 19] for applications of approach theory to hyperspaces or topological vector spaces). For a detailed motivation and more information about AP, we refer the reader to [13]. For an approach space X, we will write X for its underlying set and δX for its approach distance. In this context, Top can be proved to be concretely bicoreflectively embedded into AP, and the corresponding concrete bicoreflector T : AP → Top plays the role of forgetful functor in this setting. Recall that it was shown in [17] that the T0-objects in the sense of Marny [18], in the setting of approach spaces, are exactly those approach spaces with T0 topological coreflection. We denote by AP0 the corresponding subcategory of AP. Recently, in [10], the present authors derived a notion of symmetry for ap- proach spaces, and together with it a categorically satisfactory (i.e. sub-firmly epireflective in the sense of [6]) notion of approach bicompleteness and bicom- pletion, which has a totally different behavior compared to the behavior in the quasi-uniform case. This different behavior again highlights the structural dif- ference between approach spaces and quasi-uniform spaces: although both of them can be described using pseudo-quasi-metrics, approach spaces simultane- ously quantify topological and (pseudo-quasi)-metric spaces, so with regard to concepts such as ‘bicompleteness’ or ‘Cauchy filters’, a very different paradigm compared to the quasi-uniform one is to be expected. Let us now for the moment only recall that the category pqMet∞ of ex- tended pseudo-quasi-metric (or ∞pq-metric) spaces can be fully and concretely bicoreflectively embedded into AP, and that a T0 approach space X = (X,δX) is approach bicomplete iff the ∞pq-metric space (X,dδX ) is bicomplete in the usual sense, where dδX (x,y) := δX(x,{y}), x,y ∈ X. Functorial approach structures 93 We now want to address the question, whether or not, functorial approach structures and approach bicompleteness can be used to capture (preferably more) epireflective subcategories of Top0. 2. Results In all that follows, T : AP → Top denotes the topological bicoreflector, which in approach theory serves as the underlying functor. We refer to [13] for a detailed description of T . A functor F : Top → AP which is a section for T , i.e. for which TF = 1Top, will be called a functorial approach structure. We first need an obvious lemma characterizing all T-sections. The notation PrTop stands for the topological construct of pre-topological spaces and con- tinuous maps. For any pre-topological space, we use X for its underlying set and clX for the corresponding closure operator on X. Lemma 2.1. F : Top → AP is a section for T : AP → Top if and only if F can be written as an approach tower (Fε : Top → PrTop)ε≥0 of concrete functors with (1) F0 is the embedding of Top into PrTop, which we denote by 1Top, (2) F∞ is the indiscrete functor, which we denote by I and which equips each set with the coarsest possible topology, (3) ∀X ∈ |Top|, ∀ε ≥ 0 : FεX = ∨ ε<γ FγX, (4) ∀X ∈ |Top|, ∀ε,γ ≥ 0 : clFγX ◦ clFεX ≥ clFγ+εX. Proof. This immediately follows from the description of approach spaces in terms of towers (see [13]) and the fact that F0 = TF . � Next we show that, like in the quasi-uniform paradigm, there are “enough” functorial approach structures. The proof however becomes more intricate. Theorem 2.2. The conglomerate of T -sections is at least a proper class. Proof. Suppose that the conglomerate of all different T-sections would be in one-to-one correspondence with a set of cardinality κ. Then consider the car- dinal number 2κ > κ. Note that 2κ also is an (initial) ordinal number and that Γ := {α | α ordinal number and α < 2κ}, equipped with the inclusion ⊆ is a lattice without top element. It was proved in [12] that (Γ,⊆) has a lattice- isomorphic representation within the large lattice of bireflective subcategories 94 G. C. L. Brümmer and M. Sioen of Top (with the natural order defined there). This entails that we can find a class R of mutually different bireflective subcategories of Top which is in one-to-one correspondence with the set 2κ. (Note that we make no distinction between a bireflective subcategory and its corresponding bireflector, and that we consider such a bireflector as an endofunctor on Top). For every R ∈ R, we define: FRε :=   I ε = ∞, R ε ∈ [1,∞ [ (viewed as a functor into PrTop), 1Top ε ∈ [0, 1 [ (viewed as a functor into PrTop). Then according to Lemma 2.1, it is clear that FR := (FRε )ε≥0 is a T-section, and that {FR | R ∈ R} is a class of mutually different T-sections, being in one-to-one correspondence with the set 2κ, yielding a contradiction with the definition of κ. � Finally, we come to proving our main theorem showing that “locally around the 0-level”, all functorial approach structures however become trivial. First note that, with exactly the same proof as in the (quasi)-uniform cases treated in [7, 8, 2], we can prove that every T-section can be obtained through the spanning construction as defined by the first author. This means that given a T-section F, there exists a class A⊂ |AP| for which TA := {TA | A ∈A} is initially dense in Top and such that for all X ∈ |Top| the source (f : FX → A)A∈A,f∈Top(X,TA) is AP-initial. To summarize this we write F = 〈A〉 and we say that “A spans F”. Theorem 2.3. For every T -section F , there exists γ > 0 such that ∀ε ∈ [0,γ [ : Fε = 1Top. Proof. Take an arbitrary T-section F : Top → AP. According to Lemma 2.1, we can write F as a tower (Fε)ε≥0 of functors subject to the conditions listed there. On the other hand we know from the general spanning construction that there exists A⊂ |AP| such that TA is initially dense in Top and F = 〈A〉. In particular this means for the Sierpinski space $, that the source (f : $ → TA)A∈A, f∈Top($,TA) is initial in Top. This clearly can only happen when there exists B ∈ A such that $ can be embedded into TB as a topological subspace. This means that we can find x,y ∈ B with δB(x,{y}) = 0 and γ := δB(y,{x}) > 0. Now fix ε ∈ ] 0,γ [. Then obviously the previous implies that $ still is a pre- topological subspace of Bε := (B, tBε ) (here t B ε is the pretopological closure on Functorial approach structures 95 the level ε in the approach tower corresponding to δB, i.e. for all Y ⊂ B, tBε (Y ) := {b ∈ B | δB(b,Y ) ≤ ε}). Fix X ∈ |Top|. Because F = 〈A〉, the source (f : FX → C)C∈A,f∈Top(X,TC) is initial in AP. Therefore, FεX has to be finer than the initial pre-topological structure on X for the PrTop-source (2.2) (f : X → Cε)C∈A, f∈Top(X,TC). The initial PrTop-structure being the coarsest one on X making all functions in the source (2.2) above continuous, it certainly is coarser than clX. Because $ is a pretopological subspace of Bε, it is clear on the other hand that the initial structure for the source (2.2) above at the same time has to be finer than the initial PrTop-structure for the source (f : X → $)f∈PrTop(X,$). Because Top is fully and concretely embedded as an initially closed subcategory in PrTop, and because $ is initially dense in Top, the initial PrTop-structure for the latter source is clX. So the initial PrTop-structure for (2.2) is clX, whence FεX ≥ X and since automatically FεX ≤ F0X = X, we are done. � Let us now recall from [3] that, with the same argument as in the quasi- uniform case used in [9], it follows from the universality of AP in the sense of [18] (meaning that AP is the bireflective = initial hull of its T0-objects), which was proved in [17], that there is a one-to-one correspondence between T-sections, and sections to the functor T |AP0 : AP0 → Top0. In one direction this correspondence is simply given by restriction of T-sections to Top0. We therefore immediately have the analogue of Theorem 2.3 in this setting, providing a description of all T |AP0 -sections. This yields that we automatically also obtain: Theorem 2.4. For every T |AP0 -section F , there exists γ > 0 such that ∀ε ∈ [0,γ [ : Fε = 1Top0. Now take an arbitrary T |AP0 -section F : Top0 → AP0 and let γ > 0 be as in the theorem above. Then for all X ∈ |Top0|, the ∞pq-metric dFX only takes values in {0}∪ [γ, +∞] and therefore obviously is a bicomplete metric on X, whence FX is automati- cally approach bicomplete. This answers the question posed at the end of the first paragraph in the negative, again showing a drastically different behaviour 96 G. C. L. Brümmer and M. Sioen of the approach setting in comparison to the quasi-uniform one: the only epire- flective subcategory of Top0 we can retrieve via functorial approach structures is Top0 itself. References [1] J. Adámek, H. Herrlich and G. Strecker, Abstract and Concrete Categories (Wiley, New York, 1990). [2] G. C. L. Brümmer, Categorical aspects of the theory of quasi-uniform spaces, Rend. Istit. Mat. Univ. Trieste 30 (Suppl.) (1999), 45–74. [3] G. C. L. Brümmer, Extending constructions from the T0-spaces to all topological spaces, in preparation. [4] G. C. L. Brümmer, Functorial transitive quasi-uniformities, Categorical Topology (Proc. Conf. Toledo, Ohio, 1983), (Heldermann Verlag, Berlin, 1984), 163–184. [5] G. C. L. Brümmer, Completions of functorial topological structures, Recent Develop- ments of General Topology and its Applications (Proc. Conf. Berlin 1992), (Akademie Verlag, Berlin, 1992), 60–71. [6] G. C. L. Brümmer and E. Giuli, A categorical concept of completion of objects, Math. Univ. Carolinae 33 (1992), 131–147. [7] G. C. L. Brümmer and A. W. Hager, Completion-true functorial uniformities, Seminar- berichte Fachber. Math. Inf. FernUniv. Hagen 19 (1984), 95–104. [8] G. C. L. Brümmer and A. W. Hager, Functorial uniformization of topological spaces, Topology Appl. 27 (1987), 113–127. [9] G. C. L. Brümmer and H.-P Künzi, Bicompletion and the Samuel bicompactification, Appl. Categ. Struct. 10 (2002), 317–330. [10] G. C. L. Brümmer and M. Sioen, Approach bicompleteness and bicompletion, in prepa- ration. [11] P. Fletcher and W. F. Lindgren, Quasi-uniform Spaces, (Marcel Dekker, New York and Basel, 1982). [12] M. Hušek, Applications of category theory to uniform structures, Lecture Notes Math. 962 (Springer, Berlin, 1982), 138–144. [13] R. Lowen, Approach spaces. The Missing Link in the Topology-Uniformity-Metric Triad, (Clarendon Press, Oxford, 1997). [14] R. Lowen and M. Sioen, Proximal hypertopologies revisited, Set-Valued Analysis 6 (1998), 1–19. [15] R. Lowen and M. Sioen, Approximations in functional analysis, Results in Mathematics 37 (2000), 345–372. [16] R. Lowen and M. Sioen, Weak representations of quantified hyperspace structures, Top. Appl. 104 (2000), 169–179. [17] R. Lowen and M. Sioen, A short note on separation in AP, Appl. Gen. Top., to appear. [18] T. Marny, On epireflective subcategories of topological categories, Gen. Top. Appl. 10 (1979), 175–181. [19] M. Sioen and S. Verwulgen, Locally convex approach spaces, Appl. Gen. Top., to appear. Received January 2002 Revised December 2002 Functorial approach structures 97 Guillaume C. L. Brümmer Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa. E-mail address : gclb@maths.uct.ac.za Mark Sioen Department of Mathematics, Free University of Brussels, Pleinlaan 2, B-1020 Brussel, Belgium. E-mail address : msioen@vub.ac.be Functorial approach structures. By G. C. L. Brümmer and M. Sioen