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1.
《Quaestiones Mathematicae》2013,36(3):323-337
Abstract It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces. 相似文献
2.
《Quaestiones Mathematicae》2013,36(1-3):355-377
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. 相似文献
3.
《Quaestiones Mathematicae》2013,36(1-3):307-319
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. 相似文献
4.
《Quaestiones Mathematicae》2013,36(4):353-362
Abstract In this paper two ordered families of topological categories are studied. The first family includes the category of all abstract simplicial complexes and the subcategories of all abstract simplicial complexes of dimension less than or equal to n. The categories of the second family are bireflective subcategories of the category of all bornological spaces. All these categories are cartesian closed and have other nice properties. 相似文献
5.
6.
《Quaestiones Mathematicae》2013,36(1-3):97-106
Dense subcategories were introduced by S. Marde?i? for an inverse system approach to (categorical) shape theory. In this paper some internal characterizations of (epi,bi)dense subcategories of a topological category are given. We also show that if K ? A is a bidense subcategory then the “best approximation” of an A-object X by a K-inverse system is obtained by “modifications” of the structure of X. 相似文献
7.
《Quaestiones Mathematicae》2013,36(1-3):113-137
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:
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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;
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V carries U-initial monosources into T-initial mono-sources.
8.
A concrete category is a CCT (cartesian closed topological) extension of the category Unif of uniform spaces if 1. is cartesian closed, 2. Unif is a full, finitely productive subcategory of and the forgetful functor of extends that of Unif and 3. has initial structures. We describe the smallest CCT extension of Unif which is called the CCT hull by H. Herrlich and L.D. Nel. The objects of the CCT hull are bornological uniform spaces, i.e. uniform spaces endowed with a collection of “bounded” sets related naturally to the uniformity; the morphisms are the uniformly continuous maps which preserve the bounded sets. 相似文献
9.
《Quaestiones Mathematicae》2013,36(1-3):235-243
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 ? xU, x ? 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. 相似文献
10.
《Quaestiones Mathematicae》2013,36(1-3):227-254
Abstract Generalizing results of Herrlich and Nel, the author characterizes by means of smallest proper structures those objects X of an initially structured category for which X x—has a right adjoint, and describes the corresponding function spaces. It is shown that reduction to finally and initially dense classes is possible. The results are applied to epireflective subcategories of the category of limit spaces containing a finite non-indiscrete space, in particular to epireflective subcategories of TOP. 相似文献
11.
《Quaestiones Mathematicae》2013,36(1-3):59-84
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. 相似文献
12.
《Quaestiones Mathematicae》2013,36(1-3):23-43
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. 相似文献
13.
《Quaestiones Mathematicae》2013,36(1-3):45-57
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. 相似文献
14.
It is shown that some familiar properties of epimorphisms in the category of frames carry over to the categories of uniform and complete uniform frames. This is achieved by suitably enriching certain frame homomorphisms to uniform frame homomorphisms. 相似文献
15.
《Quaestiones Mathematicae》2013,36(3):237-253
Abstract Every topological category over an arbitrary base category X may be considered as a category of T-models with respect to some theory (i.e., functor) T from X into a category of complete lattices. Using this model-theoretic correspondence as our basic tool, we study initial and final completions of (co)fibration complete categories. For an arbitrary concrete category (A, U) over X, the process of order-theoretically completing each fibre does not usually yield an initial/final completion of (A, U). It is shown in this paper that for concrete categories which are assumed to be fibration and/or cofibration complete, initial and final completions can be constructed by completing the fibres. These completions are further shown to exhibit some interesting external properties. 相似文献
16.
In previous papers, the notions of “closedness” and “strong closedness” in set-based topological categories were introduced. In this paper, we give the characterization of closed and strongly closed subobjects of an object in the category Prord of preordered sets and show that they form appropriate closure operators which enjoy the basic properties like idempotency (weak) hereditariness, and productivity.We investigate the relationships between these closure operators and the well-known ones, the up- and down-closures. As a consequence, we characterize each of T0, T1, and T2 preordered sets and show that each of the full subcategories of each of T0, T1, T2 preordered sets is quotient-reflective in Prord. Furthermore, we give the characterization of each of pre-Hausdorff preordered sets and zero-dimensional preordered sets, and show that there is an isomorphism of the full subcategory of zero-dimensional preordered sets and the full subcategory of pre-Hausdorff preordered sets. Finally, we show that both of these subcategories are bireflective in Prord. 相似文献
17.
《Quaestiones Mathematicae》2013,36(1-3):191-205
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 Lindelöf property, and regular-closedness. With respect to the new product, a space is N-complete iff it is zerodimensional and R-complete. 相似文献
18.
《Quaestiones Mathematicae》2013,36(1-3):85-101
Abstract A synthesis of notions arising from algebraic geometry, especially those developed by Verdier in Séminaire de Géométric Algébrique IV, and the notion of topological functor (in the sense of G.C.L. Brümmer 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. 相似文献
19.
《Quaestiones Mathematicae》2013,36(3):315-339
ABSTRACT (PART II): In terms of a given Hamiltonian function the 1-form w = dH + ?j|dπj is defined, where {?j:j = 1,…, n} denotes an invariant basis of the planes of the distribution Dn. The latter is said to be canonical if w = 0 (which is analogous to the definition of Hamiltonian vector fields in symplectic geometry). This condition is equivalent to two sets of canonical equations that are expressed explicitly in term of the derivatives of H with respect to its positional arguments. The distribution Dn is said to be pseudo-Lagrangian if dπj(?j,Vh) = 0; if Dn, is both canonical and pseudo-Lagrangian it is integrable and such that H = const. on each leaf of the resulting foliation. The Cartan form associated with this construction [9] is defined a II = π2 ? ? πn. If π is closed, the distribution DN is integrable, and the exterior system {πj} admits the representation ψj = dSj in terms of a set of 0-forms Sj on M. If, in addition, the distribution DN is canonical, these functions satisfy a single first order Hamilton-Jacobi equation, and conversely. Finally, a complete figure is constructed on the basis of the assumptions that (i) the Cartan form be closed, and (ii) that the distribution Dn, be both canonical and integrable. The last of these requirements implies the existence of N functions ψA that depend on xh and N parameters wB, whose derivatives are given by ?ψA (xh, wB)/?xj = BA j (xh, ψB (xh,wB)). The complete figure then consists of two complementary foliations: the leaves of the first are described by the functions ψA and satisfy the standard Euler-Lagrange equations, while the second, that is, the transversal foliation, is represented by the aforementioned solution of the Hamilton-Jacobi equation. The entire configuration then gives rise in a natural manner to a generalized Hilbert independent integral and consequently also to a generalized Weierstrass excess function. 相似文献
20.
《Quaestiones Mathematicae》2013,36(1-4):263-280
Abstract It is shown that (concretely) Cartesian closed topological hulls can be characterized as injective hulls in a rather natural setting. The characterization of locale hulls as injective hulls in the category of (meet-) semilattices by Bruns & Lakser and Born & Kimura constitutes a special case. 相似文献