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

Abstract

We investigate the relationships among the weaker forms of the properties "Menger", "Alster" and "Lindelöf" in bitopological spaces. We give some counterexamples to show the differences between these properties. Further we introduce the weak versions of the Alster property in terms of selection principles and obtain some of their topological properties.

Games relating to weak covering properties in bitopological spaces

“…In [14] Kočinac introduced the notion of almost Menger property in topological spaces and in [9,10,18] this notion was investigated in the bitopological context. Now in this section we consider the almost Menger game on bitopological spaces and relate this notion to (i, j) -almost Menger property.…”

“…Recall that [10] a bitopological space (X, τ 1 , τ 2 ) is (i, j) -almost σ -compact if it is the union of τ j -closures of countably many τ i -compact subsets. Then we have the following theorem:…”

“…It was shown in [10] that almost σ -compact bitopological spaces are almost Alster. In this paper, after observing that the almost σ -compact property implies the property that TWO has a winning strategy in almost Menger game, it is of interest to know whether the almost Alster property and TWO has a winning strategy in the almost Menger game may be related in bitopological spaces.…”

“…On the other hand, there are quite a few studies on the weak versions of covering properties in bitopological spaces. They are mainly related to weak versions of the Menger property such as almost Menger, weakly Menger and their connections with the Menger property in bitopological spaces (see, [9,10,18]). However there is no systematic study of weak covering properties and their relations with game theory in bitopological context.…”

“…In [10] the authors characterized (i, j) -almost Alster property in terms of selection principle S .…”