Fuzzy topology on fuzzy sets: Product fuzzy topology and fuzzy topological groups

The notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy Hausdorffness, compactness, connectedness are examined. The product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group G and some properties of fuzzy topological groups are obtained.     

Uniqueness of Polish group topology

We show the limits of Mackey's theorem applied to identity sets to prove that a given group has a unique Polish group topology.Verbal sets in Abelian Polish groups and full verbal sets in the infinite symmetric group are Borel. However this is not true in general.A Polish group with a neighborhood π-base at 1 of sets from the σ-algebra of identity and verbal sets has a unique Polish group topology. It follows that compact, connected, simple Lie groups, as well as finitely generated profinite groups, have a unique Polish group topology.     

Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants.We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than .We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ^ω, for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ>κ.     

The topology of a semisimple Lie group is essentially unique

We study locally compact group topologies on simple and semisimple Lie groups. We show that the Lie group topology on such a group S is very rigid: every “abstract” isomorphism between S and a locally compact and σ-compact group Γ is automatically a homeomorphism, provided that S is absolutely simple. If S is complex, then noncontinuous field automorphisms of the complex numbers have to be considered, but that is all. We obtain similar results for semisimple groups.     

Independence of almost periodicity upon the topology of the acting group

Answering a question first explicitly stated by de Vries in 1993, we observe that for an arbitrary topological dynamical system the property of being an almost periodic point does not depend on the topology of the acting group. In other words, the traditional distinction made between the notions of an almost periodic point and of a discretely almost periodic point is unnecessary.     

A topology induced by uniformity on BL‐algebras

In this paper, we consider a collection of filters of a BL‐algebra A. We use the concept of congruence relation with respect to filters to construct a uniformity which induces a topology on A. We study the properties of this topology regarding different filters. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Abelian groups admitting a Fréchet-Urysohn pseudocompact topological group topology

We show that every Abelian group G with r₀(G)=|G|=|G|^ω admits a pseudocompact Hausdorff topological group topology T such that the space (G,T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k≥0, the Ulm-Kaplansky invariant f_p,k of G satisfies (f_p,k)^ω=f_p,k provided that f_p,k is infinite and f_p,k>f_p,i for each i>k.Our approach is based on an appropriate dense embedding of a group G into a Σ-product of circle groups or finite cyclic groups.     

The étale open topology over the fraction field of a Henselian local domain

Suppose that R is a local domain with fraction field K. If R is Henselian, then the R-adic topology over K refines the étale open topology. If R is regular, then the étale open topology over K refines the R-adic topology. In particular, the étale open topology over $L((t_1,\text{…},t_n))$ agrees with the $L[[t_1,\text{…},t_n]]$-adic topology for any field L and $n\ge 1$.     

The finest Nachbin topology is a mixed topology

Let T be a completely regular topological space. We show that the finest Nachbin topology on C(T) is the finest locally convex topology on C(T) that coincides with the compact open topology on the order bounded intervals of C(T).     

代数结构上的模糊拟伪$b$-度量

研究代数结构上的模糊(拟)伪$b$-度量. 主要结论有: (1)设$G$是一个抽象群, $\tau$是$G$上一个左不变模糊拟伪$b$-度量(伪$b$-度量)诱导的拓扑,如果$(G,\tau)$是右拓扑群,那么$(G,\tau)$是一个仿拓扑群(拓扑群); (2)设$S$是一个半群,如果$\tau$是$S$上一个不变模糊拟伪$b$-度量诱导的拓扑,那么$(S,\tau)$是一个拓扑半群.     

Order topology and bi-Scott topology on a poset

In this paper, some properties of order topology and bi-Scott topology on a poset are obtained. Order-convergence in posets is further studied. Especially, a sufficient and necessary condition for order-convergence to be topological is given for some kind of posets.     

On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for *-homomorphisms and the topology of the KK-groups which avoids entirely C^*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown-Salinas topology. We study the generalized Rørdam group , and prove that if a separable exact residually finite dimensional C^*-algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type 2^∞. In particular such an embedding exists for the C^*-algebra of a second countable amenable locally compact maximally almost periodic group.     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Advanced topology on the multiscale sequence spaces

We pursue the study of the multiscale spaces S^ν introduced by Jaffard in the context of multifractal analysis. We give the necessary and sufficient condition for S^ν to be locally p-convex, and exhibit a sequence of p-norms that defines its natural topology. The strong topological dual of S^ν is identified to another sequence space depending on ν, endowed with an inductive limit topology. As a particular case, we describe the dual of a countable intersection of Besov spaces.     

Boundaries on spacetimes: causality,topology, and group actions

The future causal boundary on a spacetime serves to explicate the causal behavior of the spacetime at future infinity. The purely causal nature of this boundary has a categorically universal nature, the category being that of chronological sets. There is an associated topology with any chronological set, replicating the appropriate topology for a spacetime. Adding the future causal boundary (and using this topology) provides a quasi-compactification. The boundary for a product spacetime can be detailed in terms of the Riemannian factor M.      

Simple density topology

We introduce a new topology on the real line generated by the simple density points for measure. We show also that a simple category density point does not lead to a new notion. Supported by research project “Analisi Reale”, Italian PRIN funds Visiting Professor at Facoltà di Economia, Università Federico II, Naples, Italy     

赋予hit—or-miss拓扑超空间动力系统的拓扑熵

设（X,d,f）为拓扑动力系统,其中X为局部紧第二可数Hausdorff空间,d为紧型度量,f为完备映射,用2^x和f分别表示由X的所有非空闭子集和所有闭子集构成的集族,（2^x,ρ,2^f）和（f,ρ,2^f）为由（X,d,f）诱导的赋予hit—or—miss拓扑的超空间动力系统．本文研究了h（X,d,f）和h（2^...     

Hyperbolic topology of normed linear spaces

In a previous paper [H. Tsuiki, Y. Hattori, Lawson topology of the space of formal balls and the hyperbolic topology of a metric space, Theoret. Comput. Sci. 405 (2008) 198–205], the authors introduced the hyperbolic topology on a metric space, which is weaker than the metric topology and naturally derived from the Lawson topology on the space of formal balls. In this paper, we characterize spaces L_p(Ω,Σ,μ) on which the hyperbolic topology induced by the norm _p coincides with the norm topology. We show the following:

(1) The hyperbolic topology and the norm topology coincide for 1<p<∞.

(2) They coincide on L₁(Ω,Σ,μ) if and only if μ(Ω)=0 or Ω has a finite partition by atoms.

(3) They coincide on L_∞(Ω,Σ,μ) if and only if μ(Ω)=0 or there is an atom in Σ.

Extremal pseudocompact Abelian groups are compact metrizable

Every pseudocompact Abelian group of uncountable weight has both a proper dense pseudocompact subgroup and a strictly finer pseudocompact group topology.

     

Group duality with the topology of precompact convergence

The Pontryagin-van Kampen (P-vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P-vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompact-open topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P-vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.