Covering properties defined by preopen sets

In this paper, we study some covering properties in topological spaces defined by preopen sets. We introduce and investigate the properties of the pre-Menger property, the almost pre-Menger property and their star versions. It is shown that the pre-Menger and the semi-Menger [Covering properties defined by semi-open sets, J. Nonlinear Sci. Appl. 9 (2016) 4388–4398] are independent properties.

On nearly Menger and nearly star-Menger spaces

In 1999, Kocinac defined and characterized the almost Menger property. Following this concept, we define and investigate nearly Menger and nearly star-Menger spaces. Every Menger space is nearly Menger, and every nearly Menger space is almost Menger. It is demonstrated that a nearly Menger space may not necessarily be a Menger space. In the similar way, we consider nearly ?-sets.

Generalized open sets and selection properties

We define and study new weak versions of the classical Menger covering property. For this we use ?-open and ?-open covers of a topological space. Relations of these properties with known weak versions of the Menger property are examined. In this way we complement the study of weak covering properties defined by selection principles.

PRODUCTS OF HUREWICZ SPACES IN THE LAVER MODEL

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel’s conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.