\(\omega\)-far points in large spaces. (Q1868878)

Let \(X\) be a crowded (i.e., no isolated points) space. A point \(p\in\beta X\setminus X\) is called an \(\omega\)-far point of \(X\) provided that \(p\not\in\text{cl}_{\beta X}D\) for any countable closed discrete set \(D\subset X\). In the paper [Colloq. Math. 41, 45--52 (1979; Zbl 0424.54012)], \textit{E. K. van Douwen} proved that normal non-Lindelöf spaces and non-compact metrizable spaces have \(\omega\)-far points, and its question of whether all non-pseudocompact crowded spaces have \(\omega\)-far points is still open. In this paper authors are interested in proving the existence of \(\omega\)-far points by constructing \(\omega\)-filters. The main result is that every non-compact Lindelöf space which is nowhere of cardinality at most \textbf{c} has an \(\omega\)-filter, and thus an \(\omega\)-far point. This provides a partial answer to the question of van Douwen.