Remote Filters and Discretely Generated Spaces

Abstract

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 ω× ω 2 with a base of order type ω 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.

Between countably compact and ω-bounded

“…A somewhat similar construction has recently been carried out in [7], however not in ZFC but using the extra assumption p = cof(M). This yields a locally compact Hausdorff space X on the underlying set (ω × C) ∪ p such that (ω × C) in its natural topology is dense open in X, the closed nowhere dense subspace p carries its ordinal topology, hence…”