Menger-type covering properties of topological spaces

Abstract

In this paper we find conditions under which the properties of Menger, almost Menger and weakly Menger are equivalent as well as the corresponding properties of Lindelöf-type. We give counterexamples that show the interrelations between those properties. The subject of our investigation is also the preservation of almost Menger and weakly Menger properties under subspaces and products. We also consider the weaker versions of Alster space and D-spaces.

Games relating to weak covering properties in bitopological spaces

“…Let (X, τ 1 , τ 2 ) be a bitopological space. We use the following notations which are similar to those used in the papers [4,13].…”

Investigations onweak versions of the Alster property in bitopological spaces and selection principles

“…Furthermore it is proven that every productively Lindelöf space of weight at most ℵ 1 is productively weakly Lindelöf. The reader is suggested to refer [4,19] for detailed informations on weakly Alster and almost Alster spaces. Now let us turn to bitopological spaces.…”

“…The following classes of covers of X will be at the center of this investigation. We will follow the similar notations as used in the papers [4,19]. G τ i : The family of all covers U of X for which each element of U is a τ i -G δ set.…”

“…Recently in [4] Babinkostova, Pansera and Scheepers have obtained some results to identify classes of spaces which are productively weakly Lindelöf and considered the weak versions of the Alster property. Additionally Kocev in [19] investigated the almost Alster spaces and proved that the product of an almost Menger and almost Alster spaces is almost Menger. In this article we study these properties in bitopological context. The idea of bitopological spaces was first initiated by J.C. Kelly in [14] and many papers have appeared in the literature so far.…”

On the Alster, Menger and D-type covering properties

“…We also give a new characterization for almost Menger spaces by neigborhood assignment. The notion of almost D-space was introduced by Kocev in [13]. A topological space (X, τ ) is an almost D-space if for every function N : X → τ such that x ∈ N(x), there exists a closed discrete subspace D of X such that x∈D N(x) = X.…”