On factorization structures and epidense hulls

Characterizations of epidense subcategories of topological categories and of existence of epidense hulls have been described in [2, 3, 4]. In this paper a similar characterization is given in a much more general setting; for example the category need not have products. The relationship between finite factorization structures and existence of epidense hulls is investigated. It is found to be analogous to the relationship between general factorization structures and epireflective hulls.     

Nuclear Typing of Frames vs Spatial Selectors

Nuclei which are defined over a class of frames are called nuclear typings. There is the dual notion of a spatial selector, and the relationship between nuclear typings and spatial selectors emanates from the duality between spatial frames and sober spaces. Especially interesting is the interplay between typings that are well-behaved with respect to certain frame quotients and selectors which similarly behave well in passage to closed sets.     

Geometric topological completions with universal final lifts

In this paper necessary and sufficient conditions are given on a concrete category over a category B so that it can be densely embedded (over B) into a geometric topological category E that admits certain universal final lifts. These conditions, as well as the class of universal final lifts, depend upon an a priori given full subcategory Δ of B. For example, E may have, depending upon Δ and B, universal coproducts or quotients or colimits. For appropriate Δ's, if B is cartesian closed then so is E.     

Maximal elements in topological and generalized metric spaces

In this note, some problems concerning existence of maximal elements in a topological as well as in a generalized metric space, equipped with an ordering, are studied. The results presented here may be considered as a partial refinement of those established in [2] for uniform structures.     

Multiepireflective subcategories

Y. Diers has defined multireflective subcategories as a generalization of reflective subcategories. In this paper, the related concepts of multiepireflective and monomultireflective subcategories are defined and investigated. It is proved that, for categories with appropriate (E,M) factorization structures, every multireflection can be expressed as the composition of an epireflection followed by a multiepireflection. Characterizations of multi-E-reflective subcategories are also given for categories with (E,M)-factorization structures. Finally, a list of subcategories of Top which are: multireflective in Top, multiepireflective in Top₂ and {initial-monosources}-multireflective in CRog T₂ is given.     

On the arc component of a locally compact Abelian group

For a topological group G, we denote by G _a the arc component of the neutral element and by the character group of G, i.e. the group of all continuous homomorphisms from G into T. We prove the following theorem: Let G be a connected locally compact abelian group and let be the embedding. Then is a topological isomorphism. In particular, the character group of the arc component of a compact abelian group is discrete. Some conclusions will be drawn.     

Open maps,colimits, and a convenient category of fibre spaces

We show that pulling back along an open map preserves all colimits in the category of weak Hausdorff k-spaces. We also show that the category of open maps over a weak Hausdorff k-space is a convenient category of fibre spaces.     

Fréchet-Urysohn fans in free topological groups

In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet-Urysohn fan S_ω in a topological group G admitting a functorial embedding [0,1]⊂G. The latter means that each autohomeomorphism of [0,1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space Q of rationals give topological groups containing S_ω.     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

ON EPIDENSE SUBCATEGORIES OF TOPOLOGICAL CATEGORIES

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.     

Ordered topological structures

The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general level of lax algebras, so that our categories do not concern just ordered topological spaces, but also sets with two interacting orders, approach spaces with an additional metric, etc.     

CATEGORIES OF TOPOLOGICAL 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.     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Injective dimension of sheaves of rational vector spaces

The Cantor–Bendixson rank of a topological space X is a measure of the complexity of the topology of X. We will be interested primarily in the case that the space is profinite: Hausdorff, compact and totally disconnected. In this paper, we prove that the injective dimension of the abelian category of sheaves of $\mathbb{Q}$-modules over a profinite space X is determined by the Cantor–Bendixson rank of X.     

The associativity equation revisited

Summary Consideration of the Associativity Equation,x (y z) = (x y) z, in the case where:I × I I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( – ,b), ( – ,b], –, +), (a, + ), or [a, + ) — whereb = 0 or –1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.     

Cohomology rings and formality properties of nilpotent groups

We analyze k-stage formality and relate resonance with this type of formality properties. For instance, we show that, for a finitely generated nilpotent group that is k-stage formal, the resonance varieties are trivial up to degree k. We also show that the cohomology ring, truncated up to degree k+1, of a finitely generated nilpotent, k-stage formal group is generated in degree 1; this criterion is necessary and sufficient for a finitely generated, 2-step nilpotent group to be k-stage formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups.     

Categorical neighborhood operators

We introduce and study a concept of neighborhood operator on a category. Such an operator is obtained by assigning a suitably axiomatized stack of subobjects - the neighborhoods - to every subobject of each object in the category. We discuss closure and interior operators, convergence, separation and compactness with respect to a neighborhood operator, defined in a natural way.     

Statistical convergence in topology

We introduce and investigate statistical convergence in topological and uniform spaces and show how this convergence can be applied to selection principles theory, function spaces and hyperspaces.     

An alternative definition of coarse structures

Roe [J. Roe, Lectures on Coarse Geometry, University Lecture Series, vol. 31, Amer. Math. Soc., Providence, RI, 2003] introduced coarse structures for arbitrary sets X by considering subsets of X×X. In this paper we introduce large scale structures on X via the notion of uniformly bounded families and we show their equivalence to coarse structures on X. That way all basic concepts of large scale geometry (asymptotic dimension, slowly oscillating functions, Higson compactification) have natural definitions and basic results from metric geometry carry over to coarse geometry.