Topological regular variation: III. Regular variation

This paper extends the topological theory of regular variation of the slowly varying case of Bingham and Ostaszewski (2010) [5] to the regularly varying functions between metric groups, viewed as normed groups (see also Bingham and Ostaszewski (2010) [6]). This employs the language of topological dynamics, especially flows and cocycles. In particular we show that regularly varying functions obey the chain rule and in the non-commutative context we characterize pairs of regularly varying functions whose product is regularly varying. The latter requires the use of a ‘differential modulus’ akin to the modulus of Haar integration.     

The Index Theorem of topological regular variation and its applications

We develop further the topological theory of regular variation of [N.H. Bingham, A.J. Ostaszewski, Topological regular variation: I. Slow variation, LSE-CDAM-2008-11]. There we established the uniform convergence theorem (UCT) in the setting of topological dynamics (i.e. with a group T acting on a homogenous space X), thereby unifying and extending the multivariate regular variation literature. Here, working with real-time topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.     

Topological regular variation: II. The fundamental theorems

This paper investigates fundamental theorems of regular variation (Uniform Convergence, Representation, and Characterization Theorems) some of which, in the classical setting of regular variation in R, rely in an essential way on the additive semigroup of natural numbers N (e.g. de Bruijn's Representation Theorem for regularly varying functions). Other such results include Goldie's direct proof of the Uniform Convergence Theorem and Seneta's version of Kendall's theorem connecting sequential definitions of regular variation with their continuous counterparts (for which see Bingham and Ostaszewski (2010) [13]). We show how to interpret these in the topological group setting established in Bingham and Ostaszewski (2010) [12] as connecting N-flow and R-flow versions of regular variation, and in so doing generalize these theorems to Rd. We also prove a flow version of the classical Characterization Theorem of regular variation.     

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hähl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).     

On an equivalence of topological vector space valued cone metric spaces and metric spaces

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. Recently this has been applied by Du (2010) [14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable, prove a completion theorem, and also obtain some more results in topological vector space valued cone normed spaces.     

Representation of Hilbert algebras and implicative semilattices

In this paper we shall give a topological representation for Hilbert algebras that extend the topological representation given by A. Diego in [4]. For implicative semilattices this representation gives a full duality. We shall also consider the representation for Boolean ring.     

Modified Bers operators and the duality of multiplicative holomorphic automorphic forms

The integral Bers operator, related to a reflection in some quasicircle, plays an important role in the theory of single-valued automorphic forms [1, 2]. Study was started in [3] of the normed spaces of measurable and multiplicative holomorphic automorphic forms for a Fuchsian group. In the present article we introduce some basic multiplicative modifications of the Bers operator and the corresponding bilinear pairing in connection with duality in the spaces of multiplicative holomorphic automorphic forms. Under study we obtain a universal norm estimate and establish selfadjointness for all operators.     

Some fixed point theorems in fuzzy reflexive Banach spaces

In this paper, we first show that there are some gaps in the fixed point theorems for fuzzy non-expansive mappings which are proved by Bag and Samanta, in [Bag T, Samanta SK. Fixed point theorems on fuzzy normed linear spaces. Inf Sci 2006;176:2910–31; Bag T, Samanta SK. Some fixed point theorems in fuzzy normed linear spaces. Inform Sci 2007;177(3):3271–89]. By introducing the notion of fuzzy and α- fuzzy reflexive Banach spaces, we obtain some results which help us to establish the correct version of fuzzy fixed point theorems. Second, by applying Theorem 3.3 of Sadeqi and Solati kia [Sadeqi I, Solati kia F. Fuzzy normed linear space and it’s topological structure. Chaos, Solitons & Fractals, in press] which says that any fuzzy normed linear space is also a topological vector space, we show that all topological version of fixed point theorems do hold in fuzzy normed linear spaces.     

Two Notes on Top ological Groups

In this short paper, we firstly give a short proof of Birkhoff-Kakutani Theorem by Moore metrizable Theorem. Then we prove that G is a topological group if it is a paratopological group which is a dense G_δ-set in a locally feebly compact regular space X.     

Representation and duality for Hilbert algebras

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.      

A class of angelic sequential non-Fréchet-Urysohn topological groups

Fréchet-Urysohn (briefly F-U) property for topological spaces is known to be highly non-multiplicative; for instance, the square of a compact F-U space is not in general Fréchet-Urysohn [P. Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980) 749-753. [27]]. Van Douwen proved that the product of a metrizable space by a Fréchet-Urysohn space may not be (even) sequential. If the second factor is a topological group this behaviour improves significantly: we have obtained (Theorem 1.6(c)) that the product of a first countable space by a F-U topological group is a F-U space. We draw some important consequences by interacting this fact with Pontryagin duality theory. The main results are the following:

(1)

If the dual group of a metrizable Abelian group is F-U, then it must be metrizable and locally compact.

(2)

Leaning on (1) we point out a big class of hemicompact sequential non-Fréchet-Urysohn groups, namely: the dual groups of metrizable separable locally quasi-convex non-locally precompact groups. The members of this class are furthermore complete, strictly angelic and locally quasi-convex.

(3)

Similar results are also obtained in the framework of locally convex spaces.

Another class of sequential non-Fréchet-Urysohn complete topological Abelian groups very different from ours is given in [E.G. Zelenyuk, I.V. Protasov, Topologies of Abelian groups, Math. USSR Izv. 37 (2) (1991) 445-460. [32]].     

Structural properties of compact groups with measure-theoretic applications

Every compact group is Baire isomorphic to a product of compact metric spaces; the isomorphism takes the Haar measure on the group to a direct product measure. This topological connection between compact groups and products of compact metric spaces provides a unified treatment for (Baire) measures on compact groups and for measures on topological products of metric spaces.     

度量投影的收敛性（英文）

讨论了赋范空间中度量投影的收敛性．得到了在局部紧集控制下，Chebyshev凸集序列的度量投影的收敛性与K－M收敛，Wijsman收敛和Kuratowski收敛都等价．本文的结论完善了M．Tsukada在[1]和[2]结果．     

Fundamental groups of compact manifolds and symmetric geometry of noncompact type

We introduce the notion of geometrical engagement for actions of semisimple Lie groups and their lattices as a concept closely related to Zimmer's topological engagement condition. Our notion is a geometrical criterion in the sense that it makes use of Riemannian distances. However, it can be used together with the foliated harmonic map techniques introduced in [8] to establish foliated geometric superrigidity results for both actions and geometric objects. In particular, we improve the applications of the main theorem in [9] to consider nonpositively curved compact manifolds (not necessarily with strictly negative curvature). We also establish topological restrictions for Riemannian manifolds whose universal cover have a suitable symmetric de Rham factor (Theorem B), as well as geometric obstructions for nonpositively curved compact manifolds to have fundamental groups isomorphic to certain groups build out of cocompact lattices in higher rank simple Lie groups (Corollary 4.5). Received: October 22, 1997     

More on remainders close to metrizable spaces

This article is a natural continuation of [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90]. As in [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90], we consider the following general question: when does a Tychonoff space X have a Hausdorff compactification with a remainder belonging to a given class of spaces? A famous classical result in this direction is the well known theorem of M. Henriksen and J. Isbell [M. Henriksen, J.R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958) 83-106].It is shown that if a non-locally compact topological group G has a compactification bG such that the remainder Y=bG?G has a Gδ-diagonal, then both G and Y are separable and metrizable spaces (Theorem 5). Several corollaries are derived from this result, in particular, this one: If a compact Hausdorff space X is first countable at least at one point, and X can be represented as the union of two complementary dense subspaces Y and Z, each of which is homeomorphic to a topological group (not necessarily the same), then X is separable and metrizable (Theorem 12). It is observed that Theorem 5 does not extend to arbitrary paratopological groups. We also establish that if a topological group G has a remainder with a point-countable base, then either G is locally compact, or G is separable and metrizable.     

Integral representation of belief measures on compact spaces

An integral representation theorem for outer continuous and inner regular belief measures on compact topological spaces is elaborated under the condition that compact sets are countable intersections of open sets (e.g. metric compact spaces). Extreme points of this set of belief measures are identified with unanimity games with compact support. Then, the Choquet integral of a real valued continuous function can be expressed as a minimum of means over the sigma-core and also as a mean of minima over the compact subsets. Similarly, for bounded measurable functions, the Choquet integral is expressed as min of means over the core, we prove in addition that it is a mean of infima over the compact subsets. Then, we obtain Choquet–Revuz' measure representation theorem and introduce the Möbius transform of a belief measure. An extension to locally compact and sigma-compact topological spaces is provided.     

The Vietoris hyperspace structure for approach spaces

We show that there exists a natural approach version of the topological Vietoris hyperspace construction [16], [17] which has several advantages over the topological version. In the first place the important structural fact that the Vietoris construction can now also be considered, not only for topological but also intrinsically for metric spaces, but equally important in the second place the fact that we can considerably strengthen fundamental classic results. In this paper we mainly pay attention to properties concerning or involving compactness. As main results, in the first place we prove that it is not merely compactness of the Vietoris hyperspace which is equivalent to compactness of the original space [3] but that actually in the approach setting the indices of compactness [7], [8], [9], [10] numerically completely coincide. In the second place the well-known result [3], [4], [15] which says that if the original space is compact metric then the Vietoris topology is metrizable by the Hausdorff metric gets strengthened in the sense that in the approach setting under the same conditions the Vietoris approach structure actually coincides with the Hausdorff metric. Classic results follow as easy corollaries. Besides these main results we also draw attention to the good functorial relationship between the Vietoris approach structures and the associated topologies.     

Metrizability of paratopological (semitopological) groups

We investigate some generalized metric space properties on paratopological (semitopological) groups and prove that a paratopological group that is quasi-metrizable by a left continuous, left-invariant quasi-metric is a topological group and give a negative answer to Ravsky?s question (Ravsky, 2001 [18, Question 3.1]). It is also shown that an uncountable paratopological group that is a closed image of a separable, locally compact metric space is a topological group. Finally, we discuss Hausdorff compactification of paratopological (semitopological) groups, give an affirmative answer to Lin and Shen?s question (Lin and Shen, 2011 [14, Question 6.9]) and improve an Arhangel?skii and Choban?s theorem. Some questions are posed.     

Some generalizations of Backʼs Theorem

The problem of the existence of jointly continuous utility functions is studied. A continuous representation theorem of Back [1] gives the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. The commodity space is locally compact and second countable. Our results generalize Back?s Theorem to non-metrizable commodity spaces with a family of not necessarily total preorders. Precisely, we consider regular commodity spaces having a weaker locally compact second countable topology.