On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

When does the Haver property imply selective screenability?

We point out that in metric spaces Haver's property is not equivalent to the property introduced by Addis and Gresham. We prove that they are equal when the space has the Hurewicz property. We prove several results about the preservation of Haver's property in products. We show that if a separable metric space has the Haver property, and the nth power has the Hurewicz property, then the nth power has the Addis-Gresham property. R. Pol showed earlier that this is not the case when the Hurewicz property is replaced by the weaker Menger property. We introduce new classes of weakly infinite dimensional spaces.     

The β-space property in monotonically normal spaces and GO-spaces

In this paper we examine the role of the β-space property (equivalently of the MCM-property) in generalized ordered (GO-)spaces and, more generally, in monotonically normal spaces. We show that a GO-space is metrizable iff it is a β-space with a Gδ-diagonal and iff it is a quasi-developable β-space. That last assertion is a corollary of a general theorem that any β-space with a σ-point-finite base must be developable. We use a theorem of Balogh and Rudin to show that any monotonically normal space that is hereditarily monotonically countably metacompact (equivalently, hereditarily a β-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a β-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos.     

A note on a set-valued extension property

Principal result: Suppose Y is metrizable. Then: (a) if X is metrizable and A⊂X is closed, then every continuous g:A→Y extends to an l.s.c. ψ:X→K(Y); (b) Y satisfies (a) for all paracompact X if and only if Y is completely metrizable.     

The three space problem in topological groups

We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p=c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups.     

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω₁, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2^x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.     

QN-spaces, wQN-spaces and covering properties

The main results of the paper are as follows: covering characterizations of wQN-spaces, covering characterizations of QN-spaces and a theorem saying that Cp(X) has the Arkhangel'ski?ˇ property (α₁) provided that X is a QN-space. The latter statement solves a problem posed by M. Scheepers [M. Scheepers, Cp(X) and Arhangel'ski?ˇ's αi-spaces, Topology Appl. 89 (1998) 265-275] and for Tychonoff spaces was independently proved by M. Sakai [M. Sakai, The sequence selection properties of Cp(X), Preprint, April 25, 2006]. As the most interesting result we consider the equivalence that a normal topological space X is a wQN-space if and only if X has the property S₁(Γshr,Γ). Moreover we show that X is a QN-space if and only if Cp(X) has the property (α₀), and for perfectly normal spaces, if and only if X has the covering property (β₃).     

CLP-compactness for topological spaces and groups

We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct:

(i)

a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and

(ii)

a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G→H, with a Hausdorff group H, are open).

     

o-Boundedness of free topological groups

Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F(X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle fin_?(O,Ω) (the latter means that for every sequence 〈un〉_n∈ω of open covers of T there exists a sequence 〈vn〉_n∈ω such that vn∈[un]^<ω and for every F∈[X]^<ω there exists n∈ω with F⊂?vn). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.     

Function spaces over GO-spaces: Part I

We describe the structure of spaces of continuous step functions over GO-spaces. We establish a relation between the Dedekind completion of a GO-space L and properties of the space of continuous functions from L to 2 with finitely many steps. We use the established relation to prove that a countably compact GO-space L has Lindelöf Cp(L) iff the Dedekind remainder of L is Lindelöf and every compact subspace of L is metrizable. Or equivalently, a countably compact GO-space L has Lindelöf Cp(L) iff every compact subspace of L is metrizable and a Gδ-set in L. Other results are obtained.     

Building suitable sets for locally compact groups by means of continuous selections

If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1-2) (1990) 181-194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.     

Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems

Hyperspace dynamical system (E²,f²) induced by a given dynamical system (E,f) has been recently investigated regarding topological mixing, weak mixing and transitivity that characterize orbit structure. However, the Vietoris topology on E² employed in these studies is non-metrizable when E is not compact metrizable, e.g., E=Rn. Consequently, metric related dynamical concepts of (E²,f²) such as sensitivity on initial conditions and metric-based entropy, could not even be defined. Moreover, a condition on (E²,f²) equivalent to the transitivity of (E,f) has not been established in the literature. On the other hand, Hausdorff locally compact second countable spaces (HLCSC) appear naturally in dynamics. When E is HLCSC, the hit-or-miss topology on E² is again HLCSC, thus metrizable. In this paper, the concepts of co-compact mixing, co-compact weak mixing and co-compact transitivity are introduced for dynamical systems. For any HLCSC system (E,f), these three conditions on (E,f) are respectively equivalent to mixing, weak mixing and transitivity on (E²,f²) (hit-or-miss topology equipped). Other noticeable properties of co-compact mixing, co-compact weak mixing and co-compact transitivity such as invariants for topological conjugacy, as well as their relations to mixing, weak mixing and transitivity, are also explored.     

More generalizations of pseudocompactness

We introduce a covering notion depending on two cardinals, which we call O-[μ,λ]-compactness, and which encompasses both pseudocompactness and many other known generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to O-[ω,ω]-compactness.We provide several characterizations of O-[μ,λ]-compactness, and we discuss its connection with D-pseudocompactness, for D an ultrafilter. The connection turns out to be rather strict when the above notions are considered with respect to products. In passing, we provide some conditions equivalent to D-pseudocompactness.Finally, we show that our methods provide a unified treatment both for O-[μ,λ]-compactness and for [μ,λ]-compactness.     

Universal free G-spaces

For a compact Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free G-spaces. We show that such a universal free G-space cannot be compact.     

On one-point metrizable extensions of locally compact metrizable spaces

For a non-compact metrizable space X, let E(X) be the set of all one-point metrizable extensions of X, and when X is locally compact, let EK(X) denote the set of all locally compact elements of E(X) and be the order-anti-isomorphism (onto its image) defined in [M. Henriksen, L. Janos, R.G. Woods, Properties of one-point completions of a non-compact metrizable space, Comment. Math. Univ. Carolin. 46 (2005) 105-123; in short HJW]. By definition λ(Y)=_?n<ωclβX(Un∩X)\X, where Y=X∪{p}∈E(X) and {Un}_n<ω is an open base at p in Y. We characterize the elements of the image of λ as exactly those non-empty zero-sets of βX which miss X, and the elements of the image of EK(X) under λ, as those which are moreover clopen in βX\X. This answers a question of [HJW]. We then study the relation between E(X) and EK(X) and their order structures, and introduce a subset ES(X) of E(X). We conclude with some theorems on the cardinality of the sets E(X) and EK(X), and some open questions.     

A note on unbounded strongly measure zero subgroups of the Baer-Specker group

We show that it is consistent with ZFC that there exist:

(1)

An unbounded (with respect to _?∗) and strongly measure zero subgroup of ZN, but without the Menger property.

(2)

An unbounded (with respect to _?∗) and strongly measure zero subgroup of ZN with the Menger property which does not have the Rothberger property.

This answers the last two problems which remained from a classification project of Machura and Tsaban.     

On the Delta set and catenary degree of Krull monoids with infinite cyclic divisor class group

Let M be a Krull monoid with divisor class group Z, and let S⊆Z denote the set of divisor classes of M which contain prime divisors. We find conditions on S equivalent to the finiteness of both Δ(M), the Delta set of M, and c(M), the catenary degree of M. In the finite case, we obtain explicit upper bounds on maxΔ(M) and c(M). Our methods generalize and complement a previous result concerning the elasticity of M.