The Josefson–Nissenzweig theorem and filters on ω

For a free filter F on ω ω , endow the space NF=ω ∪ \ N F = ω ∪ { p F } , where pF ∈ ω p F ∉ ω , with the topology in which every element of ω ω is isolated whereas all open neighborhoods of pF p F are of the form A∪ \ A ∪ { p F } for A∈ F A ∈ F . Spaces of the form NF 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 NF N F carries a sequence μ ₙ:n∈ ω ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that μ ₙ(f)→ 0 μ n ( f ) → 0 for every bounded continuous real-valued function f on NF N F if and only if F^*≤ KZ F ∗ ≤ K Z , that is, the dual ideal F^* F ∗ is Katětov below the asymptotic density ideal Z Z . Consequently, we get that if F^*≤ KZ F ∗ ≤ K Z , then: (1) if X is a Tychonoff space and NF N F is homeomorphic to a subspace of X , then the space Cₚ^*(X) C p ∗ ( X ) of bounded continuous real-valued functions on X contains a complemented copy of the space c₀ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and NF 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.