Abstract
For a free filter F on ω , endow the space NF = ω ∪ {pF} , where pF ∉ ω , with the topology in which every element of ω is isolated whereas all open neighborhoods of pF are of the form A ∪ {pF} for A ∈ F . Spaces of the form NF 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 NF carries a sequence ⟨μn:n ∈ ω⟩ of normalized finitely supported signed measures such that μn(f) → 0 for every bounded continuous real-valued function f on NF 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 NF is homeomorphic to a subspace of X, then the space Cp∗(X) of bounded continuous real-valued functions on X contains a complemented copy of the space c0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and NF 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.