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].     

Versions of separability in bitopological spaces

We study selective versions of separability in bitopological spaces. In particular, we investigate these properties in function spaces endowed with the topology of pointwise convergence and the compact-open topology.     

On pairwise Lindelöf bitopological spaces

In this paper, we shall continue the study of bitopological separation axioms begun by Kelly and obtained some results. Furthermore, we introduce two concepts of pairwise Lindelöf bitopological spaces and the properties for them are established. We also show that a pairwise Lindelöf space is not hereditary property.     

A characterization of the Menger property by means of ultrafilter convergence

We characterize various Menger/Rothberger-related properties, and discuss their behavior with respect to products.     

Measurable cardinals and the cardinality of Lindelöf spaces

We obtain from the consistency of the existence of a measurable cardinal the consistency of “small” upper bounds on the cardinality of a large class of Lindelöf spaces whose singletons are Gδ sets.     

Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or D

We discuss relationships in Lindelöf spaces among the properties “Menger”, “Hurewicz”, “Alster”, “productive”, and “D”.     

Pathologies in cohomology of non-paracompact Hausdorff spaces

We construct a non-paracompact Hausdorff space for which ?ech cohomology does not coincide with sheaf cohomology. Moreover, the sheaf of continuous real-valued functions is neither soft nor acyclic, and our space admits non-numerable principal bundles.     

A selective bitopological version of the Reznichenko property in function spaces

For a Tychonoff space X we consider the compact-open and the topology of pointwise convergence on the set of all continuous real-valued functions, define a selective version of the Reznichenko property connecting the two topologies and characterize it dually via a suitable covering property of the space X.     

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.     

Indestructibility of compact spaces

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to _ω1${\omega }_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba?s “_ℵ1${\aleph }_1$-Borel Conjecture” is equiconsistent with the existence of an inaccessible cardinal.     

The metacompactness of spaces with bases of subinfinite rank

Let X be a set. A collection $\text{U}$of subsets of X has subinfinite rank if whenever $\text{V}$ ? $\text{U}$, ∩$\text{V}$≠ø, and $\text{V}$ is infinite, then there are two distinct elements of $\text{V}$, one of which is a subset of the other. Theorem. AT₁space with a base of subinfinite rank is hereditarily metacompact.     

Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts

We characterize metric spaces X whose hyperspaces X² (or Bd(X)) of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.     

Rothberger's property in all finite powers

A space X has the Rothberger property in all finite powers if, and only if, its collection of ω-covers has Ramseyan properties.     

More stratifiable function spaces

If X is a compact-covering image of a closed subspace of product of a σ-compact Polish space and a compact space, then Ck(X,M), the space of continuous maps of X into M with the compact-open topology, is stratifiable for any metric space M.If X is σ-compact Polish, K is compact and M metric then every point of Ck(X×K,M) has a closure-preserving local base, and hence this function space is M₁.     

Locally arcwise connected metrizable spaces with the fixed point property are complete-metrizable

Metrizable spaces which have no closed subset homeomorphic to the real halfline are considered. Complete-metrizability of such locally arcwise connected spaces is proved (Corollary 4) and a simple formula for a complete metric is given in Theorem 3.     

On D-Paracompact spaces

Following Pareek a topological space X is called D-paracompact if for every open cover $\text{A}$ of X there exists a continuous mapping f from X onto a developable T₁-space Y and an open cover $\text{B}$ of Y such that { f^-1[B]|B ∈ $\text{B}$ } refines $\text{A}$. It is shown that a space is D-paracompact if and only if it is subparacompact and D-expandable. Moreover, it is proved that D-paracompactness coincides with a covering property, called dissectability, which was introduced by the author in order to obtain a base characterization of developable spaces.     

The bicompletion of the Hausdorff quasi-uniformity

We study conditions under which the Hausdorff quasi-uniformity UH of a quasi-uniform space (X,U) on the set P₀(X) of the nonempty subsets of X is bicomplete.Indeed we present an explicit method to construct the bicompletion of the T₀-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform T₀-spaces (X,U) for which the Hausdorff quasi-uniformity of their bicompletion on is bicomplete.     

Products of Menger spaces: A combinatorial approach

We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in almost all canonical models of set theory of the real line. On the other hand, we establish productive properties for versions of Menger's property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger's property is strictly between Menger's and Hurewicz's classic properties. We include a number of open problems emerging from this study.     

Filter spaces

The category FIL of filter spaces and cauchy maps is a topological universe. This paper establishes the foundation for a completion theory forT ₂ filter spaces.     

On productively Lindelöf spaces

The class of spaces such that their product with every Lindelöf space is Lindelöf is not well-understood. We prove a number of new results concerning such productively Lindelöf spaces with some extra property, mainly assuming the Continuum Hypothesis.