On the topological structure of the Levi-Civita field

Two topologies on the Levi-Civita field R will be studied: the valuation topology induced by the order on the field, and another weaker topology induced by a family of seminorms, which we will call weak topology. We show that each of the two topologies results from a metric on R, that the valuation topology is not a vector topology while the weak topology is, and that R is complete in the valuation topology while it is not in the weak topology. Then the properties of both topologies will be studied in details; in particular, we give simple characterizations of open, closed, and compact sets in both topologies.     

Properties of topologies of information retrieval systems

This paper studies topological properties of different topologies that are possible on the space of documents as they are induced by queries in a query space together with a similarity function between queries and documents. The main topologies studied here are the retrieval topology (introduced by Everett and Cater) and the similarity topology (introduced by Egghe and Rousseau).The studied properties are the separation properties T₀, T₁, and T₂ (Hausdorff), proximity and connectedness. Full characterizations are given for the diverse topologies to be T₀, T₁, or T₂. It is shown that the retrieval topology is not necessarily a proximity space, while the similarity topology and the pseudo-metric topology always are proximity spaces. A characterization of connectedness in terms of the Boolean NOT-operator is given, hereby showing the intimate relationship between IR and topology.     

Some topologies on a Šerstnev probabilistic normed space

In this paper we will introduce two other topologies, coarser than the so-called strong topology, on a class of Šerstnev probabilistic normed spaces, and obtain some important properties of these topologies. We will show that under the first topology, denoted by τ₀, our probabilistic normed space is decomposable into the topological direct sum of a normable subspace and the subspace of probably null elements. Under the second topology, which is in fact the inductive limit topology of a family of locally convex topologies, the dual space becomes a locally convex topological vector space.     

Function spaces and product topologies on powers of spaces

A study is made of two classes of product topologies on powers of spaces: the general box product topologies, and the general uniform product topologies. Some examples are given and some results are shown about the properties of these general product spaces. This is applied to show that certain spaces of continuous functions with the fine topology are homeomorphic to box product spaces, and certain spaces of continuous functions with the uniform topology are homeomorphic to uniform product spaces.     

偏序集上的测度拓扑与稠拓扑

In this paper,the concepts of the essential topology and the density topology of dcpos are generalized to the setting of general posets.Basic properties of the essential topology and relations with other intrinsic topologies are explored.Comparisons between the density topology and the measurement topology are made.Via the essential topology,the density topology and the measurement topology,we obtain properties and characterizations of bases of continuous posets.We also provide some new conditions for a continuous poset to be an algebraic poset.     

Sequences of capacities,with connections to large-deviation theory

Acapacity is a set function with some regularity properties on a Hausdorff spaceE. Many measures and all sup measures are examples. The set of capacities onE can be endowed with two natural topologies. The narrow topology corresponds to the weak topology for probability measures, while the vague topology corresponds to the vague topology for Radon measures. The connection between these topologies and large-deviation principles was noted in recent joint work with W. Vervaat. Here, the theory of capacities and their topologies is developed in directions which have implications for large-deviation theory.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada. Much of it was done while the author enjoyed the kind hospitality of the Delft University of Technology.     

Strict and Mixed Topologies on Function Spaces

In this paper we prove that the strict topology on spaces of continuous and holomorphic functions on a BANACH space can be considered a mixed topology. Using this fact, we obtain new results about the strict topology, as an application of the general properties of mixed topologies.     

Preorderable topologies and order-representability of topological spaces

We furnish characterizations of topologies that coincide with the lower topology or with the order topology of some total preorder defined on a set. Leaning on these characterizations we introduce some applications to the study of continuous and semicontinuous order-representability properties of topological spaces.     

On abstract and almost-abstract density topologies

We concentrate on topologies generated by lower and almost-lower density operators on measurable spaces. Among others the existence of the smallest in the sense of inclusion abstract density topology on measurable space is investigated. Moreover, the separation axioms for such topologies are studied.     

Hypertopologies on function spaces

The aim of this paper is to continue Naimpally’s seminal papers [16], [17], [18], i.e. we investigate topological properties of spaces which force the coincidence of convergences of functions associated with different hyperspace topologies. For example a metric spaceX is locally compact iff the topological convergence and the convergence induced by the Fell topology coincide onC(X,IR). Moreover, the proximal topology on the space of functions, not necessarily continuous, is studied in great detail.     

On hyperspace topologies via distance functionals in quasi-metric spaces

Summary Let (X,d) be a quasi-pseudo-metric space. We investigate hyperspace topologies on P₀(X) defined by distance functionals. In particular, the K-topology is introduced and compared with other hyperspace topologies. Some properties of the Wijsman topology and the K-topology are explored.     

Gruppentopologien auf nichtabelschen abzählbaren Gruppen

Group topologies of countable groups are constructed. They form a dense subset of the lattice of all group topologies. Every such group topology induces an injective mapping from the set of all filters of the natural numbers, which are finer than the filter of those sets of natural numbers having finite complements, into the lattice of group topologies. A sufficient condition is given, which groups can be topologized. It is shown, that every group can be topologized by a non-linear group topology, if it can be topologized by a group topology.     

Effective dynamics for chaos synchronization in networks with time-varying topology

A coupled map lattice whose topology changes at each time step is studied. We show that the transversal dynamics of the synchronization manifold can be analyzed by the introduction of effective dynamical quantities. These quantities are defined as weighted averages over all possible topologies. We demonstrate that an ensemble of short time observations can be used to predict the long-term behavior of the lattice. Finally, we point out that it is possible to obtain a lattice with constant topology in which the dynamical behavior is asymptotically identical to one of the time-varying topology.     

Topologies of uniform convergence. The property in the sense of Arens-Dugundji and the sequential property

We study properties of topologies τ _sup and τ _inf, which are, respectively, the supremum and infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. The main result of the research consists in obtaining necessary and sufficient conditions for the properness and admissibility in the sense of Arens-Dugundji for the topology τ _inf. In this paper, we introduce the notion of a sequentially proper topology and establish necessary and sufficient conditions for the sequential properness of the topology τ _inf. We also consider a special case when the maximal proper topology and the maximal sequentially proper topology coincide on the set C(X, Y).     

A note on Golomb topologies

In 1959 Golomb defined a connected topology on ?. An analogous Golomb topology on an arbitrary integral domain was defined first by Knopfmacher-Porubský [KP97] and then again in a recent work of Clark [Cl17]. Here we study properties of Golomb topologies.     

Representable topologies and locally connected spaces

The objective of this paper is to study certain order-theoretic properties of locally connected topologies. In the main theorem we prove that the assertion that a locally connected topology which satisfies the countable chain condition has the continuous representability property is undecidable in ZFC set theory. Some related problems and generalizations are also included.     

Locally Convex Product and Direct Sum Cones

In Motallebi and Saiflu [3], products and direct sums of cones have been investigated based on convex quasi-uniform structures. In this paper, we define the notion of product and direct sum cone topologies based on a suitable order relation and discuss the duality properties. Using the polars of neighborhoods, we define some of the X-topologies which are isomorphic to each other. In the condition of the strict separation property, these topologies will be isomorphic to a product (or direct sum) cone topology. This leads us to study weak-topologies, topologies of pointwise convergence and the finest X-topologies.     

Topologies of the Hyperspace which Are Induced by Extension Spaces

The hyperspace of all nonvoid closed subsets of a topological space will be topologized by means of different methods; each of them generalizes known definitions of special hyperspace topologies. One method uses the VIETORIS topology in the hyperspaces of certain extension spaces. Another one uses certain systems of closed subsets and leads to just the same class of hyperspace topologies. In natural order VIETORIS topology is the supremum of this class, and for locally compact spaces FLACHSMEYER topology is the infimum of this class.     

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)?{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.