1,721,038 research outputs found
On the minimality of powers of minimal omega-bounded abelian groups
No full textAbstract. We describe the structure of totally disconnected minimal ω- bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the p-adic integers Zp. In this case the description is obtained by means of a functorial correspondence, based on Pontryagin duality, between topological and linearly topologized groups introduced by Tonolo. As an application we answer the question (posed in Pseudocompact and countably compact abelian groups: Cartesian products and minimality, Trans. Amer. Math. Soc. 335 (1993), 775–790) when arbitrary powers of minimal ω-bounded abelian groups are minimal. We prove that the positive answer to this question is equivalent to non-existence of measurable cardinals
Uniformly Approchable Functions and Spaces
No full textUniformly approachable (UA) functions are a common generalization of uniformly continuous functions an d perfect functions. We study UA-functions and UA-spaces i. e. those uniform spaces in which every real valued continuous function is UA. Such spaces properly include the UC-spaces (Atsuji spaces). We characterize the weakly-UA subspaces of the real line and give a new characterization of the UC spaces. We prove a topological result which implies, under the continuum hypothesis, the existence of a subset M of the the n-dimensional euclidean space R^n such that if two continuous functions f, g from R^n to R are are not constant on any open set and g(M) is a subset of f(M), then f=g
ON THE LATTICE OF LINEAR MODULE TOPOLOGIES
No full textWe study various features of the lattice of linear
module topologies on a module and their impact on the structure of the abstract module.
A particular emphasis is given to: a) permanence properties of -minimal topologies
(=atoms in the subset of of Hausdorff topologies) in analogy with the
theory of minimal topological groups; b) the usual equivalence between linear topologies and
the equivalence classes of the atoms; c) characterization of the abstract modules such
that certain classes in are singletons (in particular, modules such that each
non-discrete linear topology is topologically artinian). Point c) involves the class of modules
having all proper quotients artinian
Finiteness of topological entropy for locally compact abelian groups
No full textWe study the locally compact abelian groups in the class , that is, having only continuous endomorphisms of finite topological entropy, and in its subclass , that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian -groups. We show that locally compact abelian -groups of finite rank belong to , and that those of them that belong to are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian -groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem
Algebraic entropy for Abelian groups
No full textThe theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. In this paper the algebraic entropy, introduced in 1965 by Adler, Konheim and McAndrew, is studied. The so-called Addition Theorem is proved. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all whose endomorphisms have zero algebraic entropy. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is also proved
A characterization of the maximally almost periodic abelian groups
Full text linkWe introduce a categorical closure operator g in the category of topological abelian groups (and continuous homomorphisms) as a Galois closure with respect to an appropriate Galois correspondence defined by means of the Pontryagin dual of the underlying group. We prove that a topological abelian group G is maximally almost periodic if and only if every cyclic subgroup of G is g-closed. This generalizes a property characterizing the circle group from (Studia Sci. Math. Hungar. 38 (2001) 97-113. A characterization of the circle group and the p-adic integers via sequential limit laws, preprint). and answers an appropriate version of a question posed in (A characterization of the circle group and the p-adic integers via sequential limit laws, preprint)
Ore localization of amenable monoid actions and applications towards entropy addition formulas and the bridge theorem
Full text linkFor a left action of a cancellative
right amenable monoid on a discrete Abelian group , we construct its Ore
localization , where is the
group of left fractions of ; analogously, for a right action
on a compact space , we construct its Ore
colocalization . Both constructions
preserve entropy, i.e., for the algebraic entropy and for
the topological entropy one has
and
, respectively.
Exploiting these constructions and the theory of quasi-tilings, we extend the
Addition Theorem for , known for right actions of countable
amenable groups on compact metrizable groups, to right actions
of cancellative right amenable monoids
(with no restrictions on the cardinality) on arbitrary compact groups .
When the compact group is Abelian, we prove that
coincides with , where
is the dual left action on the
discrete Pontryagin dual , that is, a so-called Bridge Theorem. From
the Addition Theorem for and the Bridge Theorem, we obtain
an Addition Theorem for for left actions
on discrete Abelian groups, so far known
only under the hypotheses that either is torsion or is locally
monotileable.
The proofs substantially use the unified approach towards entropy based on
the entropy of actions of cancellative right amenable monoids on appropriately
defined normed monoids
TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS
No full textIt 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
An additivity theorem for uniformly continuous functions
No full textWe consider metric spaces X with the nice property that any continuous function f:X → R which is uniformly continuous on each set of a finite cover of X by closed sets, is itself uniformly continuous. We characterize the spaces with this property within the ample class of all locally connected metric spaces. It turns out that they coincide with the uniformly locally connected spaces, so they include, for instance, all topological vector spaces. On the other hand, in the class of all totally disconnected spaces, these spaces coincide with the UC spaces
- …
