The Josefson–Nissenzweig theorem and filters on ω

For a free filter F on ω, endow the space N _F=ω∪{p _F}, where p _F∉ω, with the topology in which every element of ω is isolated whereas all open neighborhoods of p _F are of the form A∪{p _F} for A∈F. Spaces of the form N _F constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space N _F carries a sequence ⟨μ _n:n∈ω⟩ of normalized finitely supported signed measures such that μ _n(f)→0 for every bounded continuous real-valued function f on N _F if and only if F ^∗≤ _KZ, that is, the dual ideal F ^∗ is Katětov below the asymptotic density ideal Z. Consequently, we get that if F ^∗≤ _KZ, then: (1) if X is a Tychonoff space and N _F is homeomorphic to a subspace of X, then the space C _p ^∗(X) of bounded continuous real-valued functions on X contains a complemented copy of the space c ₀ endowed with the pointwise topology, (2) if K is a compact Hausdorff space and N _F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.