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The Cartesian Closed Topological Hull of the Category of Completely CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES H. L. BENTLEY E. LOWEN-COLEBUNDERS The cartesian closed topological hull of the category of completely regular filterspaces Cahiers de topologie et géométrie différentielle catégoriques, tome 33, no 4 (1992), p. 345-360 <http://www.numdam.org/item?id=CTGDC_1992__33_4_345_0> © Andrée C. Ehresmann et les auteurs, 1992, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE VOL. XXXIII-4 (1992) ET GÉOMÉTRIE DIFF-DRENTIELLE CA TÉGORIQUES THE CARTESIAN CLOSED TOPOLOGICAL HULL OF THE CATEGORY OF COMPLETELY REGULAR FILTERSPACES by H. L. BENTLEY and E. LOWEN-COLEBUNDERS RESUME. Dans cet article, on construit 1’enveloppe topo- logique cart6sienne ferm6e de la cat6gorie de tous les es- paces fibres completement r6guliers, et des applications uniform6ment continues. Les objets de cette nouvelle categorie sont caracterises : ce sont les espaces filtres pseudotopologiques p-r6guliers, a domaine p-ferm6. 1. INTRODUCTION. It is a well known fact that Creg, the category of com- pletely regular topological spaces and continuous maps, is not cartesian closed and hence is inconvenient for many purposes in homotopy theory, topological algebra and functional analysis. Fortunately Creg can be fully embedded in a cartesian closed topological hull. This hull is known to be the category Cemb of c-embedded convergence spaces and continuous maps. The ob- jects of Cemb have been internally characterized as those con- vergence spaces that are w-regular, w-closed domained and pseudotopological [191. The category Cemb has proved to be extremely useful in the study of function spaces of realvalued maps 161. For a discussion of all the nice aspects of a cartesian closed category we refer to the original paper of H. Herrlich [131 and to a survey paper of L. Nel 1211. The role of cartesian clo- sedness in functional analysis is clearly demonstrated in [11]. Complete regularity is one of the most interesting notions in topology, since it is closely related to the real number sys- tem. In order to develop a theory about completely regular ex- tensions of topological spaces, Bentley, Herrlich &#x26; Ori introdu- ced a notion of complete regularity for merotopic spaces [4]. In one of their main results it is stated that a merotopic space is a subspace in Mer (the category of merotopic spaces and uni- formly continuous maps) of a completely regular topological space if and only if it is a completely regular filterspace. From this theorem, it follows that the category CregFil of 345 completely regular filterspaces plays an important role when studying extensions or completions. However, like its topological counterpoint, CregFil is not cartesian closed. Therefore, with regard to some particular problems related to the areas mentio- ned above, it has some deficiencies. It is well known that the larger category Fil of all filtermerotopic spaces is a cartesian closed supercategory of CregFil. In some respects, it is too big to retain enough structure of CregFil. Hence it seems desirable to find a cartesian closed topological hull of CregFil. The con- struction of this hull and the internal characterization of its ob- jects are the main subjects of this paper. The construction of this cartesian closed topological hull is a new example in the list of "improvement by enlargement". Other examples are Cemb, the cartesian closed topological hull of Creg, the category Pstop of pseudotopological spaces which is the cartesian closed topo- logical hull of the category Prtop of pretopological spaces, and the category Ant of Antoine spaces which is the cartesian closed topological hull of Top. A survey on the situation of Top and Prtop can be found in [1S]. 2. PRELIMINARIES. In order to keep the exposition as brief as possible we assume familiarity with merotopic spaces (see e.g. 114, 17, 2, 20]). We recall the basic definitions. A merotopic space is a set X together with a subset y of p2X such that: (M1) If A corefines 1) B and A E Y , then BEy. (M2) V XEX, on a2) .vey. (M3) {O} E Y and 0%y. (M4) If AUBEY then AEY or BEY. The members of y are said to be micromeric. y is called the merotop..v. The explicit mention of the merotopy is often sup- pressed. A map f: X-Y between merotopic spaces X and Y is cal- led uniformly continuous iff whenever A is micromeric in X, then f (A) = {f(A)|A E A) is micromeric in Y. Mer is the concrete category of merotopic spaces and uniformly continuous maps. Mer is a topological construct. A 1) A corefines B iff VAEA3BEB with B C A. We denote AB or B&#x3E;A. 2) ac = {FCX|XeF}. 346 collection B of subsets of a merotopic space X is called a uniform cover of X if whenever A is micromeric in X there is a BEB and AEA such that A CB. X is called a filter(mero topic) space if it is a merotopic space and every micromeric collection is corefined by a micro- meric filter. Micromeric filters are also called Cauchy filters. It is clear that a filterspace is completely determined by its Cauchy filters. The full subcategory of Mer whose objects are the fil- terspaces is denoted by Fil. Fil is bicoreflective in Mer and Fil is cartesian closed. A Cauch.y space is a filterspace which satisfies: If A is micromeric and B is micromeric with O E A^B^. then AV B is mi- cromeric, where: A^B=AnB|AEA, BEB}, AVB={AEA, BEB}. A merotopic space is said to be completel)’ regular 141 if whenever A is a micromeric collection in X, then so is the collection {BCX| A is completely within B for some A EA}. That A is completely iiithin B means that there exists a uni- formly map f:X-&#x3E;[0,1] with f(A)C{0} and f(X|B)C{1}. Here [0.1] is understood to carry its usual topological structure. The category of all symmetric (=R0 ) convergence spaces and then a fortiori also the category of all symmetric topologi- cal spaces can be fully embedded in Fil. The embedding functor F maps a symmetric convergence space X to the filter space FX where A is micromeric in FX iff there is a filter F, convergent in X and corefining A. We will identify the category of symme- tric convergence spaces (symmetric topological spaces) with its isomorphic image through F and call this category ConvS (Tops respectively). The objects in Convs (or Tops) will simply be called convergence spaces (or topological spaces). In particular the real line with its usual topology will be considered as a topological space and also as a filterspace. For both the notation R will be used. Thus X is a convergence space iff it is a filterspace sa- tisfying the following additional properties: - If A and B are micromeric in X and if for some point -V E X we have AVX micromeric and BVX micromeric, then AV B is mi- 347 cromeric. - For every micromeric collection A in X there exists ve X such that AVX is micromeric. In any merotopic space X we will use the short notation X A X-&#x3E; A or A-&#x3E; A for the expression "AVX is micromeric on X". We say that A converges to X. Note that the convergence spa- ces we are dealing with satisfy Fisher’s axioms E10L Also, in our context, a convergence space is automatically symmetric. It means that a filter F converges to X whenever FV x! is convergent. This will cause no problems when applying results from the literature on convergence spaces (without im- plicit symmetry) since we will mainly be interested in c-em- bedded convergence spaces. and they are always symmetric. In 181 Bourdaud described the cartesian closed topological hull of the category of completely regular topological spaces. In order to give an internal characterization of the objects of the hull, he used the following notions: A convergence space X is pseudotopological 191 if a filter F converges to -Y whenever all ultrafilters finer than F converge to .B. A convergence space X is w-regular 1221 if clwxA conver- ges to x whenever A converges to X. Here where c1wX is the closure operator of the initial topology aX determined by the source (f:X-&#x3E;R)fE Hom(X.R) In view of the isomorphism of the category of symmetric convergence spaces and Convs we may use both terminologies for the functions in Hom(X,R): we may call them continuous (between the convergence spaces X and R) or uniformly conti- nuous (between X and R, considered as filterspaces). A convergence space X is w-closed domained [81 if for every filter F on X the set is WX-closed. Note that, even without implicit assumption, this property implies symmetry. Bourdaud has shown that the category of a-regular, (v-clo- sed domained pseudotopological convergence spaces is a carte- sian closed hull of Creg. This hull is also called the category of c-embedded spaces. Other characterizations of c-embedded spaces can be found in the literature, see e.g. [6]. In most papers on this subject however the Hausdorff property is impli- 348 citly assumed.
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