Combinatorial properties of filters and open covers for sets of real numbers

We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set script X sign of real numbers have the covering property of Menger if, and only if, each filter on ω associated with its countable ω-cover is a P⁺ filter.