# Showing papers in "Quaestiones Mathematicae in 1977"

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TL;DR: In this article, conditions are established under which a given class of objects in a topological category will have a Cartesian closed coreflective hull, and the main theorem is used to discover new Cartesian-closed topological categories and to unify a diversity of known special results.

Abstract: Conditions are established under which a given class of objects in a topological category will have a Cartesian closed coreflective hull. The main theorem is used to discover new Cartesian closed topological categories and to unify a diversity of known special results. It also provides a mild criterion for the existence of Cartesian closed topological hulls.

31 citations

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TL;DR: In this paper, the relation between connectedness and disconnection in topological contexts is discussed, and a discussion of the relationship between relatedness and disconnection is presented in the context of topology.

Abstract: (1977). RELATIVE CONNECTEDNESSES AND DISCONNECTEDNESSES IN TOPOLOGICAL CATEGORIES. Quaestiones Mathematicae: Vol. 2, No. 1-3, pp. 297-306.

26 citations

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TL;DR: A survey of recent results concerning categorical constructions on topological groups can be found in this paper, with particular emphasis on free topology groups and coproducts of these groups.

Abstract: This paper is a survey of recent (and some not so recent, results concerning categorical constructions on topological groups, with particular emphasis on free topological groups and coproducts (free products) of topological groups. An extensive bibliography is included.

15 citations

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TL;DR: In this paper, a concept of normality for nearness spaces is introduced which agrees with the usual normality in the case of topological spaces, is hereditary, and is preserved under the taking of the nearness completion.

Abstract: A concept of normality for nearness spaces is introduced which agrees with the usual normality in the case of topological spaces, is hereditary, and is preserved under the taking of the nearness completion. It is proved that the nearness product of a regular contigual space and a normal nearness space is always normal. The locally fine nearness spaces are studied, particularly in relation to normality conditions.

14 citations

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TL;DR: In this paper, the authors characterise these functors in terms of the unique extension of structure functors defined on the subcategory of separated objects (of the domain category), which leads to a solution of some problems due to G.R.-E.C. Brummer [1,2] and L.L. Skula's characterization of the bireflective subcategories of Top.

Abstract: R.-E. Hoffmann [5,6] has introduced the notion of an (E,M)-universally topological functor, which provides a categorical characterization of the T0-separation axiom of general topology. In this paper, we characterise these functors in terms of the unique extension of structure functors defined on the subcategory of “separated” objects (of the domain category). This, in turn, leads to a solution of some problems due to G.C.L. Brummer [1,2]. Other results include a generalization of L. Skula's characterization of the bireflective subcategories of Top [10].

12 citations

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TL;DR: In this article, it was shown that these sheaves are reflective in the corresponding category of presheaves and that the resulting reflection is stalk preserving (Proposition 2) and that for paracompact spaces X, they are exactly those BAN-sheaves S such that each SU, U open.

Abstract: This paper presents a number of results concerning sheaves on a topological space, with values in the category BAN of Banach spaces, over K = R or O, and linear contractions. After showing that these sheaves are reflective in the corresponding category of presheaves (Proposition 1) and that the resulting reflection is stalk preserving (Proposition 2), we concentrate on the approximation sheaves, these being BAN-sheaves satisfying a strong patching condition originally due to Auspitz [1]. The interest in these particular sheaves lies in the fact that they are precisely the BAN-sheaves arising as sheaves of continuous sections of the appropriate kind of Banach fibre spaces [1] and thus central to the representation of Banach spaces by continuous sections. Here, we show that the approximation sheaves on any space are characterized as the BAN-presheaves injective relative to certain maps (Proposition 3) and that, for paracompact spaces X, they are exactly those BAN-sheaves S such that each SU, U open...

12 citations

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TL;DR: In this article, the authors restate some theorems of P. Fletcher and W.F. Lindgren on transitive quasi-uniformities and S. Salbany [Thesis, Univ. Cape Town, 1971] on compactification and completion.

Abstract: This paper is motivated by the search for natural extensions of classical uniform space results to quasi-uniform spaces. As instances of such extensions we restate some theorems of P. Fletcher and W.F. Lindgren [Pacific J. Math. 43 (1971), 619–6311 on transitive quasi-uniformities and of S. Salbany [Thesis, Univ. Cape Town, 1971] on compactification and completion. The theorems as restated describe properties of certain right inverses of the functor which forgets the quasi-uniform structure and retains one induced topology (for Fletcher and Lindgren's work), respectively retains both induced topologies (for Salbany's work). Accordingly we investigate systematically the process by which the right inverses of the forgetful functors can be extended from the classical setting to one of these settings, and from one of these to the other.

10 citations

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TL;DR: In this paper, a general Galois correspondence between classes of monomorphisms and injective objects is established, which generalizes at the same time the correspondence between monomorphism and injectivity.

Abstract: First a general Galois correspondence is established, which generalizes at the same time the correspondence between classes of monomorphisms and injective objects and the correspondence between cla...

9 citations

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TL;DR: In this article, a purely topological construction of such a factorisation is given, in which the right factor is the class of spreads and the left factor has a certain property hereditarily.

Abstract: With the introduction of several new factorisation theorems, this paper is intended to show that previous efforts of the authors [3] [5] and of Strecker [15] to describe the factorisations involving connectedness are incomplete. In Section 1 we give a purely topological construction of such a factorisation, in which the right factor is the class of spreads and the left factor has a certain property hereditarily: crucially, not all members of the left factor need be quotients. Section 2 shows that, given a left factor consisting of onto maps in the category T of topological spaces, then the class of mappings with the relevant properties hereditarily is also a left factor, and the result of section 1 is a particular case of this. Section 3 combines the material in [3] on intrinsic connexion properties with ideas of Preuss (see [1]) on disconnectednesses to yield another range of factorisations, for example, involving the maps with strongly connected fibres; and Section 4 notes some outstanding prob...

8 citations

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TL;DR: In this article, a new product for topological spaces is defined by means of interior covers, which preserves paracompactness, zero-dimensionality (in the covering sense), the Lindelof property, and regular closedness.

Abstract: Examples are provided which demonstrate that in many cases topological products do not behave as they should. A new product for topological spaces is defined in a natural way by means of interior covers. In general this is no longer a topological space but can be interpreted as categorical product in a category larger than Top. For compact spaces the new product coincides with the old. There is a converse: For symmetric topological spaces X the following conditions are equivalent: (1) X is compact; (2) for each cardinal k the old and the new product Xk coincide; (3) for each compact Hausdorff space Y the old and the new product X x Y coincide. The new product preserves paracompactness, zero-dimensionality (in the covering sense), the Lindelof property, and regular-closedness. With respect to the new product, a space is N-complete iff it is zerodimensional and R-complete.

7 citations

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TL;DR: In this article, Hong et al. considered a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource) topological and fibre small.

Abstract: Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied: if h: UA → UB and g: VA → VB are morphisms with Gh = Tg, there exists a morphism f: A → B such that Uf = h and Vf = g; V carries U-initial monosources into T-initial mono-sources. The functor V has many nice properties which shed light on the blending of the “topology” and “algebra”; e.g., V is a topologically algebraic functor in the sense of Y.H. Hong. An ([Etilde],[Mtilde]) version of O. Wyler's “Taut Lift Theorem” is used to show that the existence of a left adjoint to V is related to Condition (ii). It is also shown that certain topological algebraic reflections arise as Topological Algebraic Situations from algebraic and topological surjective reflections.

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TL;DR: The relationship between Wallman's construction of a compact T1-space and Flachsmeyer's inverse limit spaces of inverse systems of decomposition spaces is investigated in this paper.

Abstract: The relationship between Wallman's construction of a compact T1-space [9] and Flachsmeyer's inverse limit spaces of inverse systems of decomposition spaces [2] is investigated. There are connections between Wallman spaces and inverse limits, which were initiated by Alexandroff in 1928. Some old theorems using inverse limits have shorter proofs now. On the other hand we obtain a new method to treat Wallman compactifications in terms of inverse limit spaces. A suitable notion in this context is the “prime-filter space”, having an interesting maximality property. This space seems to be proper to examine prime ideals in C(X).

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TL;DR: In this article, the authors survey results and open questions about categories of T1-spaces on which the wallman compactification induces an epireflection and prove results on spaces whose wallman remainder is Hausdorff.

Abstract: The first part of this paper surveys results and open questions about categories of T1-spaces on which the wallman compactification induces an epireflection. The second part proves results on spaces whose Wallman remainder is Hausdorff.

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TL;DR: In this paper, it is shown that the reflective subcategory of objects injective relative to the functor part of the monad is reflective if and only if there exists a free monad associated to the unad.

Abstract: It is well known that there is a one to one correspondence between idempotent monads in a category and reflective subcategories. In this paper it is examined what replaces the reflective subcategory if the idempotent monad is replaced (a) by a monad and (b) by a symmetric unad. It is shown that in case (a) one obtains the weakly reflective subcategory of objects injective relative to the functor part of the monad. In case (b) one obtains a proto-reflection and it is shown that (for complete categories) the associated orthogonal subcategory is reflective if and only if there exists a free monad associated to the unad.

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TL;DR: In this article, the Arbib-Manes approach to machines in categories is used to construct the categories Mach(C) of machines and TBeh(C), where c is a class of state-behaviour processes.

Abstract: Following the Arbib-Manes approach to machines in categories, the categories Mach(C) of machines and TBeh(C) of total behaviours are constructed, where c is a class of state-behaviour processes. It is shown that the total external behaviour functor E: Mach(C) → TBeh(C) has a left adjoint and that the free realization of any total behaviour is reachable. Furthermore, the restriction of E to the full subcategory of Mach(C) with all reachable machines as objects has as right adjoint the minimal realization functor.

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TL;DR: In this article, the authors extended some results about equi-uniformly continuous families of functions to the general setting of equimorphic families in a category, motivated by their previous work on proximally fine and on equi p-fine uniform spaces.

Abstract: Motivated by his previous work on proximally fine and on equi-p-fine uniform spaces, the author extends some results about equi-uniformly continuous families of functions to the general setting of equi-morphic families in a category

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TL;DR: In this paper, a notion of completeness of uniform spaces is introduced, which corresponds to a set of cannectors meeting the conditions of uniformity, and the notion of compactness of topological spaces is a special case of the completeness.

Abstract: A connector U on a space S is a function from S to the power set of S such that each x in s belongs to its image. The image of x is denoted by xU. In other words, the relation {(x,y): y ϵ xU, x ϵ S) is a reflexive binary relation. A space with a certain set of connectors is a generalization of topological spaces as well as uniform spaces. In this paper, a notion of completeness of such a space is introduced. This completeness corresponds to completeness of uniform spaces if a set of cannectors meets the conditions of uniformity. Compactness of topological Spaces is a special case of the completeness.

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TL;DR: In this article, the graphic extension of a mapping f: X → Y and an extension e of X → [Xtilde] of X is defined, and the restriction of the projection Π: X Y → Y to the closure of the graph of f in [X tilde] X Y is defined.

Abstract: Given a mapping f: X → Y and an extension e: X → [Xtilde] of X, the restriction of the projection Π: [Xtilde] X Y → Y to the closure of the graph of f in [Xtilde] X Y is called the graphic extension of f with respect to e. It is shown that this approach is widely applicable to various types of topological extensions of mappings found in the literature and often gives simpler proofs of their existence, properties, and results relating to them.

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TL;DR: In this article, it is shown that there is an extension of Mather's result when the range is only assumed to be a compact semianalytic set of some real Euclidean space, which is an obvious maximal candidate for which computations can be carried out using only classical polynomial algebra.

Abstract: John Mather has proved that infinitesimal stability implies stability for proper maps in the category of smooth manifolds. This result gives a computable algebraic criterion for stability. In this paper it is shown that there is an extension of Mather's result when the range is only assumed to be a compact semianalytic set of some real Euclidean space—this class of spaces is an obvious maximal candidate for which computations can be carried out using only classical polynomial algebra. The proof depends on a splitting theorem for the restriction map from the smooth functions on a Euclidean space to those on a closed subset and is proved by an algebraic-geometric method derived from the work of B. Malgrange. No knowledge of functional analysis is assumed although an alternative analytic method for proving the main result is also indicated. Only simple applications are given (mostly to functions defined locally in the neighbourhood of an isolated hypersurface singularity of the type studied by J. Mi...

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TL;DR: In this paper, it was shown that the forgetful functor from the category of contiguity spaces to the class of generalized proximity spaces is topological, and that the right adjoint right inverse of this functor extends the inverse of the forgetfulness functor.

Abstract: It is shown that the forgetful functor from the category of contiguity spaces to the category of generalized proximity spaces is topological, and that the right adjoint right inverse of this functor extends the inverse of the forgetful functor from the category of totally bounded uniform spaces to the category of proximity spaces.

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TL;DR: In this article, it was shown that all subsets of the pre-nearness space axioms lie in a lattice of bireflections or bicoreflections.

Abstract: The framework in which nearness spaces were defined by H. Herrlich [1] and [2], leads one to consider the supercategory Pow of the category Near of nearness spaces, having as objects all pairs (X,ξ), where X is a set and ξ ⊂ P(P(X)) is any subset of the power set of the power set of X, and as morphisms f: (X,ξ) → (Y,n) all functions f: X → Y such that, if A ϵ ξ then fA □ {f(A) | A ξ A} ϵ η. In this paper we show that the full subcategories of Pow comprising the objects satisfying subsets of the prenearness space axioms lie in a lattice of bireflections or bicoreflections. This serves as a first step towards the aim of characterizing all bireflective (resp. bicoreflective) and even all initially complete subcategories of Pow.

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TL;DR: In this article, the authors characterize those shape morphisms from X to Y which are induced by continuous maps between X and Y, where Y is a paracompact Hausdorff space.

Abstract: Let Y be a paracompact Hausdorff space. We characterize those shape morphisms from X to Y which are induced by continuous maps from X to Y.

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TL;DR: A synthesis of notions arising from algebraic geometry, especially those developed by Verdier in Seminaire de Geometric Algebrique IV, and the notion of topological functor (in the sense of G.C.L. Brummer and R.-E. Hoffmann) is made in this article.

Abstract: A synthesis of notions arising from algebraic geometry, especially those developed by Verdier in Seminaire de Geometric Algebrique IV, and the notion of topological functor (in the sense of G.C.L. Brummer and R.-E. Hoffmann) is made. In particular, Grothendieck topologies are shown to be topological over the category of categories with pullbacks and pullback preserving functors, and consequences derived.

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TL;DR: In this article, the authors present some recent results obtained by Romanian mathematicians in the field of general and categorical topology and present some current research results obtained in what may be called the topological study of a category.

Abstract: The purpose of this paper is twofold: first, to present some recent results obtained by Romanian mathematicians in the field of general and categorical topology; second, to present some current research results obtained by the author in what may be called the topological study of a category. Accordingly, the paper is divided into two parts. The author wishes to express his gratitude to the organizing committee of this Symposium for the kind invitation to present' this paper.

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TL;DR: In this paper, a semi-adjoint functor with a left co-unadjointness is presented, which can be factored through a category of semad algebras.

Abstract: If a functor U has a left co-unadjoint then U can be factored through a category of semad algebras. An analogue of the Beck monadicity theory is obtained. If R is a ring without a left unit but satisfying R2 = R then the category of unitary left R-modules need not be monadic over Set. The forgetful functor has, however, a left co-unadjoint for which a comparison functor is an equivalence of categories. Another example of a semadic functor is obtained by composing the forgetful functor from Abelian groups to Set with the doubling functor. The semi-adjoint situations in the senses of Medvedev and Davis are examined.