Square of countably compact groups without non-trivial convergent sequences

We show in ZFC that the existence of a countably compact Abelian group without non-trivial convergent sequences implies the existence of a countably compact group whose square is not countably compact.This improves a result obtained by van Douwen in 1980: the existence of a countably compact Boolean group without non-trivial convergent sequences implies the existence of two countably compact groups whose product is not countably compact in ZFC.Hart and van Mill showed in 1991 the existence of a countably compact group whose square is not countably compact under Martin's Axiom for countable posets. We show that the existence of such an example does not depend on some form of Martin's Axiom.     

On Pseudocompact, Countably Compact, Locally Bicompact Mappings, and k-Mappings

We consider various definitions of a pseudocompact mapping and the basic properties of pseudocompact mappings. Moreover, we consider the definition of countable compactness of a continuous mapping and study the properties of a countably compact mapping similar to the corresponding properties for countably compact spaces and also the interrelation between countable compactness and pseudocompactness of mappings. We also extend the notions of local bicompactness and k-space to continuous mappings.     

Sequential closure in the space of measures

We show that there is a compact topological space carrying a measure which is not a weak^? limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of arbitrarily high levels of complexity in this sequential hierarchy. It follows that there is a compact space in which the sequential closure cannot be obtained in countably many steps. However, we show that this is not the case for our spaces where the sequential closure is always obtained in countably many steps.     

On measures on Rosenthal compacta

We show that if K is Rosenthal compact which can be represented by functions with countably many discontinuities then every Radon measure on K is countably determined. We also present an alternative proof of the result stating that every Radon measure on an arbitrary Rosenthal compactum is of countable type. Our approach is based on some caliber-type properties of measures, parameterized by separable metrizable spaces.     

The weak convergence of uniform measures

This paper presents a necessary and sufficient condition for the weak convergence of uniform measures on an arbitrary Hausdorff uniform space in terms of their projections in metric spaces. This result was inspired by and extends a result of Bartoszynski which characterizes the weak convergence of countably additive measures on C[0,1] in terms of their projections in finite dimensional spaces.     

On characterization and properties of replete Wallman spaces

The subset of lattice regular zero-one valued measures on an algebra generated by a lattice (a Wallman-type space) which integrates all lattice continuous functions on an arbitrary setX is introduced and some properties of it are presented.Then repleteness of certain Wallman spaces is considered and used in finally establishing conditions whereby the space of lattice regular zero-one valued measures on the algebra generated by a lattice which are countably additive (a Wallman-type space) is realcompact.     

Countably compact groups from a selective ultrafilter

We prove that the existence of a selective ultrafilter on implies the existence of a countably compact group without non-trivial convergent sequences all of whose powers are countably compact. Hence, by using a selective ultrafilter on , it is possible to construct two countably compact groups without non-trivial convergent sequences whose product is not countably compact.

     

A remark on CD 0(K)-spaces

A representation of the CD ₀(K)-space is given in [1, 2] for a compact Hausdorff space K without isolated points. We generalize this to an arbitrary countably compact space K without any assumption on isolated points.     

Compactoid filters and USCO maps

The aim of this paper is to report in a short and self-contained way on the properties of compactoid and countably compactoid filters. We apply them to some questions in both topology and analysis such as the generation and extension of USCO maps, the study of some properties of K-analytic spaces and the study of bounds for the weight of compact sets in spaces obtained through inductive operations.     

Dixmier's theorem for sequentially order continuous Baire measures on compact spaces

We prove that a Baire measure (or a regular Borel measure) on a compact Hausdorff space is sequentially order continuous as a linear functional on the Banach space of all continuous functions if and only if it vanishes on meager Baire subsets, a result parallel to a much earlier theorem of Dixmier. We also give some results on the relation between sequentially order continuous measures on compact spaces and countably additive measures on Boolean algebras.

     

Maximal pseudocompact spaces and the Preiss-Simon property

We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.     

A generalization of Krasnoselskii’s eigenvalue theorem

We give a generalization of Krasnoselskii’s eigenvalue theorem to countably condensing set-valued maps in Banach spaces, where the method is to use a fixed point theorem for compact maps. This is based on the fact that there is a compact fundamental set for a countably condensing map.     

Insertion of semi-continuous functions and sequences of sets

We investigate the relations between decreasing sequences of sets and the insertion of semi-continuous functions, and give some characterizations of countably metacompact spaces, countably paracompact spaces, monotonically countably paracompact spaces (MCP), monotonically countably metacompact spaces (MCM), perfectly normal spaces and stratifiable spaces.     

A normal countably compact not absolutely countably compact space

We construct an example of a normal countably compact not absolutely countably compact space. We also prove that every hereditarily normal countably compact space is absolutely countably compact and suggest a method for construction of hereditarily normal spaces without property .

     

Projective limits of complex measures and martingale convergence

In this paper we prove, for signed or complex Radon measures on completely regular spaces, the analogue of Prokhorov's criterion on the existence of the projective limit of a compatible system of measures. Because of loss of mass under projections this cannot be reduced to the case of positive measures. Countable projective limits are, as in the case of positive measures, particularly simple, the sole condition now being the boundedness of the total variations. It is shown, with the help of the martingale convergence theorem, that the densities of these complex measures with respect to their variations, converge in an appropriate sense. This work is part of an extended project on the mathematical theory of path integrals. Received: 18 June 1997 / Revised version: 11 August 2000/?Published online: 9 March 2001     

Countable compactness, hereditary π-character, and the Continuum Hypothesis

We prove that the Continuum Hypothesis is consistent with the statement that countably compact regular spaces that are hereditarily of countable π-character are either compact or contain an uncountable free sequence. As a corollary we solve a well-known open question by showing that the existence of a compact S-space of size greater than ℵ₁ does not follow from the Continuum Hypothesis.     

Countably compact topological group topologies on free Abelian groups from selective ultrafilters

Tkachenko showed in 1990 the existence of a countably compact group topology on the free Abelian group of size c using CH. Koszmider, Tomita and Watson showed in 2000 the existence of a countably compact group topology on the free Abelian group of size c² using a forcing model in which CH holds.Wallace's question from 1955, asks whether every both-sided cancellative countably compact semigroup is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. In 1996, Robbie and Svetlichny constructed a Wallace semigroup under CH. In the same year, Tomita constructed a Wallace semigroup from MAcountable.In this note, we show that the examples of Tkachenko, Robbie and Svetlichny, and Koszmider, Tomita and Watson can be obtained using a family of selective ultrafilters. As a corollary, the constructions presented here are compatible with the total failure of Martin's Axiom.     

On minimal strongly KC-spaces

In this article we introduce the notion of strongly KC-spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space (X, τ) is maximal countably compact if and only if it is minimal strongly KC, and apply this result to study some properties of minimal strongly KC-spaces, some of which are not possessed by minimal KC-spaces. We also give a positive answer to a question proposed by O.T. Alas and R.G. Wilson, who asked whether every countably compact KC-space of cardinality less than c has the FDS-property. Using this we obtain a characterization of Katítov strongly KC-spaces and finally, we generalize one result of Alas and Wilson on Katìtov-KC spaces. This research was supported by NSFC of China (No. 10671173).     

Strong Minkowski Separation and Co-Drop Property

In the framework of topological vector spaces, we give a characterization of strong Minkowski separation, introduced by Cheng, et al., in terms of convex body separation. From this, several results on strong Minkowski separation are deduced. Using the results, we prove a drop theorem involving weakly countably compact sets in locally convex spaces. Moreover, we introduce the notion of the co-drop property and show that every weakly countably compact set has the co-drop property. If the underlying locally convex space is quasi-complete, then a bounded weakly closed set has the co-drop property if and only if it is weakly countably compact.