Some observations on the Baireness of C(X) for a locally compact space X

Abstract

We prove some consistency results concerning the Moving Off Property for locally compact spaces, and thus the question of whether their function spaces are Baire. IntroductionThe Moving Off Property was introduced in [11] to characterize when C k (X) satisfies the Baire Category Theorem, for q-spaces X. Here we shall only be concerned with locally compact spaces (which are q), and so won't define q. We shall assume all spaces are Hausdorff. Definition.A moving off collection for a space X is a collection K of nonempty compact sets such that for each compact L, there is a K ∈ K disjoint from L. A space satisfies the Moving Off Property (MOP) if each moving off collection includes an infinite subcollection with a discrete open expansion.

Normality Versus Paracompactness in Locally Compact Spaces

“…Theorem 5.17 [41]. If a Hausdorff space Z is locally countable, locally compact, and closed subspaces of ≤ 2 ℵ 0 have the MOP, then Z has the MOP.…”

“…MM(S)[S] is also relevant for questions concerning the Baireness of C k (X), for locally compact X (see [21,30,41]).…”

“…It follows that closed subspaces of X of size ≤ 2 ℵ 0 are locally compact and metrizable, so satisfy the MOP by 5.6. On the other hand, Theorem 5.17 [41]. If a Hausdorff space Z is locally countable, locally compact, and closed subspaces of ≤ 2 ℵ 0 have the MOP, then Z has the MOP.…”

“…For the purposes of 5.3, however, we get ℵ 1 -collectionwise Hausdorff just from the Souslin forcing, since the space is first countable. MM(S)[S] is also relevant for questions concerning the Baireness of C k (X), for locally compact X (see [21,30,41]).…”