Remote filters and discretely generated spaces

Rodrigo Hernández-Gutiérrez

Alas, Junqueira and Wilson asked whether there is a discretely generated locally compact space whose one point compactification is not discretely generated and gave a consistent example using CH. Their construction uses a remote filter in $\omega\times{}^{\omega}2$ with a base of order type $\omega_1$ when ordered modulo compact subsets. In this paper we study the existence and preservation (under forcing extension) of similar types of filters, mainly using small uncountable cardinals. With these results we show that the CH example can be constructed in more general situations.